Waveguides and scattering devices incorporating epsilon-negative and/or mu-negative slabs

ABSTRACT

Waveguides and scattering devices are made from a pair of slabs, at least one slab being either an “epsilon-negative (ENG)” layer in which the real part of permittivity is assumed to be negative while its permeability has positive real part, or a “mu-negative (MNG)” layer that has the real part of its permeability negative but its permittivity has positive real part. The juxtaposition and pairing of such ENG and MNG slabs under certain conditions lead to some unusual features, such as resonance, complete tunneling, zero reflection and transparency. Such materials also may be configured to provide guided modes in a waveguide having special features such as mono-modality in thick waveguides and the presence of TE modes with no cut-off thickness in thin parallel-plate waveguides. Using equivalent transmission-line models, the conditions for the resonance, complete tunneling and transparency are described as well as the field behavior in these resonant paired structures. A “matched” lossless ENG-MNG pair is configured to provide “ideal” image displacement and image reconstruction.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present patent application claims priority from U.S. ProvisionalPatent Application Nos. 60/474,976, filed Jun. 2, 2003, and 60/475,028,filed Jun. 2, 2003. The contents of these patent applications are herebyincorporated by reference in their entireties.

FIELD OF THE INVENTION

The present invention relates to waveguides and scattering devices madefrom multiple layers of electromagnetic complex materials with at leastone material having negative real permittivity or negative realpermeability.

BACKGROUND OF THE INVENTION

In 1967, Veselago postulated theoretically an electromagnetic complexmaterial in which both permittivity and permeability were assumed tohave negative real values, and he analyzed plane wave propagation insuch a medium, which he called a “left-handed (LH)” medium [V. G.Veselago, “The electrodynamics of substances with simultaneouslynegative values of ε and μ,” Soviet Physics Uspekhi, Vol. 10, No. 4, pp.509–514, 1968 (in Russian Usp. Fiz. Nauk, Vol. 92, pp. 517–526, 1967].According to Veselago's analysis, in such a “double-negative (DNG)”material [R. W. Ziolkowski, and E. Heyman, “Wave propagation in mediahaving negative permittivity and permeability,” Phys. Rev. E., Vol. 64,No. 5, 056625, 2001], the Poynting vector of a plane wave isantiparallel with its phase velocity. In recent years, Shelby et al. inan article entitled “Experimental verification of a negative index ofrefraction,” Science, vol. 292, no. 5514, pp. 77–79, 2001, inspired bythe work of Pendry et al. in articles entitled “Magnetism fromconductors and enhanced nonlinear phenomena,” IEEE Trans. MicrowaveTheory Tech, Vol. 47, No. 11, pp. 2075–2081, November 1999, and“Low-frequency plasmons in thin wire structures,” J. of Physics:Condensed Matter, Vol. 10, pp. 4785–4809, 1998, reported theconstruction of a composite medium in the microwave regime by arrangingarrays of small metallic wires and split ring resonators. The anomalousrefraction for this medium was demonstrated in the afore-mentionedShelby et al. article and in the following articles: D. R. Smith, etal., “Composite medium with simultaneously negative permeability andpermittivity,” Phys. Rev. Lett., Vol 84, No. 18, pp. 4184–4187, 2000;and R. A. Shelby, et al., “Microwave transmission through atwo-dimensional, isotropic, left-handed metamaterial,” Appl. Phys.Lett., Vol. 78, No. 4, pp. 489–491, 2001. Various aspects of this classof metamaterials are now being studied by several groups worldwide, andmany ideas and suggestions for potential applications of these mediahave been mentioned.

For example, the present inventors have suggested the possibility ofhaving thin, sub-wavelength cavity resonators in which a layer of the“double-negative” (DNG) medium is paired with a layer of conventionalmaterial (i.e., a “double-positive (DPS)” medium) in articles entitled:N. Engheta, “An idea for thin subwavelength cavity resonators usingmetamaterials with negative permittivity and permeability,” IEEEAntennas and Wireless Propagation Lett., Vol. 1, No. 1, pp. 10–13, 2002;N. Engheta, “Guided waves in paired dielectric-metamaterial withnegative permittivity and permeability layers” URSI Digest, USNC-URSINational Radio Science Meeting, Boulder, Colo., Jan. 9–12, 2002, p. 66;and N. Engheta, “Ideas for potential applications of metamaterials withnegative permittivity and permeability,” Advances in Electromagnetics ofComplex Media and Metamaterials, NATO Science Series, (editors S.Zouhdi, A. H. Sihvola, M. Arsalane), Kluwer Academic Publishers, pp.19–37, 2002. As explained in these articles, theoretical results haverevealed that a slab of DNG metamaterial can act as a phasecompensator/conjugator. Thus, by combining such a slab with another slabmade of a conventional dielectric material one can, in principle, have a1-D cavity resonator whose dispersion relation does not depend on thesum of thicknesses of the interior materials filling this cavity, butinstead it depends on the ratio of these thicknesses. The inventorslater extended this work to the analyses of parallel-plate waveguidescontaining a pair of DPS and DNG layers, guided modes in open DNG slabwaveguides, and mode coupling between open DNG and DPS slab waveguides.In each of these problems, the inventors found that when a DNG layer iscombined with, or is in proximity of, a DPS layer interesting andunusual properties are observed for wave propagation within thisstructure. Indeed, the paired DNG-DPS bilayer structures were found toexhibit even more interesting properties than a single DNG or DPSslab—properties that are unique to the wave interaction between the DNGand DPS layers.

By exploiting the anti-parallel nature of the phase velocity andPoynting vectors in a DNG slab, the present inventors theoreticallyfound the possibility of resonant modes in electrically thinparallel-plate structures containing such DNG-DPS bilayer structures.Following those works, a first set of preliminary results and ideas forthe guided modes in a parallel-plate waveguide containing a pair of DNGand DPS slabs was presented by the present inventors. Later, in anarticle entitled “Anomalous mode coupling in guided-wave structurescontaining metamaterials with negative permittivity and permeability,”Proc. 2002 IEEE-Nanotechnology, Washington D.C., Aug. 26–28, 2002, pp.233–234., the present inventors showed the effects of the anomalous modecoupling between DNG and DPS open waveguides located parallel to, and inproximity of, each other. Some other research groups have also exploredcertain aspects of waveguides involving DNG media.

Most of the work in the area of metamaterials reported in the recentliterature has so far been concerned with the wave interaction with DNGmedia, either by themselves or in juxtaposition with conventional (DPS)media. However, as will be explained in more detail below, the presentinventors have now recognized that “single-negative (SNG)” materials inwhich only one of the material parameters, not both, has a negative realvalue may also possess interesting properties when they are paired in aconjugate manner. These media include the epsilon-negative (ENG) media,in which the real part of permittivity is negative but the realpermeability is positive, and the mu-negative (MNG) media, in which thereal part of permeability is negative but the real permittivity ispositive. For instance, the idea of using such a combination to providean effective group velocity that would be antiparallel with theeffective phase velocity, and thus acting as an effective left-handed(LH) medium has been explored by Fredkin et al. in “Effectiveleft-handed (negative index) composite material,” Appl. Phys. Lett.,Vol. 81, No. 10, pp. 1753–1755, 2 Sept. 2002.

The present application addresses the characteristics of ENG-MNG bilayerstructures, such characteristics including resonance, completetunneling, transparency, and guided modes. Such bilayer structures arethe subject of the present invention.

SUMMARY OF THE INVENTION

The present invention provides devices such as waveguides filled withpairs of parallel or concentric layers where the first layer is made upof either (a) a material with negative real permittivity but positivereal permeability (ENG) or (b) a material with negative realpermeability but positive real permittivity (MNG) and where the secondlayer is made up of either (c) a material with both negative realpermittivity and permeability (DNG), (d) a conventional material withboth positive real permittivity and permeability (DPS), or the ENG orMNG material not used in the first layer, in a given range of frequency.Waveguides in accordance with the invention have special features suchas mono-modality in thick waveguides and the presence of transverseelectric (TE) modes with no cut-off thickness in thin parallel-platewaveguides. On the other hand, scattering devices in accordance with theinvention recognize the transverse magnetic (TM) or transverse electric(TE) wave interaction with a pair of ENG-MNG slabs and juxtapose andpair such ENG and MNG slabs so as to lead to some unusual features suchas resonance, zero reflection, complete tunneling and transparency. Theinvention takes note of the field distributions inside and outside suchpaired slabs, including the reflection and transmission from this pair,and the flow of the Poynting vector in such structures when thezero-reflection conditions are satisfied. The equivalenttransmission-line models with appropriate distributed series and shuntreactive elements are derived and applied in order to derive thenecessary and sufficient conditions for zero reflection, resonance,complete tunneling and transparency, and to explain the seeminglyanomalous field behavior in these paired structures. The inventionparticularly illustrates that pairing the ENG and MNG slabs may exhibitresonance phenomenon, and that such a resonance is one of the reasonsbehind the transparency for these paired slabs and the unusual fieldbehavior within them. Furthermore, the inventors have discussed severalcharacteristics of the tunneling conditions, such as the roles of thematerial parameters, slab thicknesses, dissipation, and angle ofincidence. Finally, as a potential application of the conjugate matchedlossless pair of ENG-MNG slabs, the inventors have proposed an idea foran “ideal” image displacement and image reconstruction utilizing such apair.

BRIEF DESCRIPTION OF THE DRAWINGS

The characteristic features of the present invention will be apparentfrom the following detailed description of the invention taken inconjunction with the accompanying drawings, of which:

FIG. 1 a illustrates the geometry of a parallel-plate waveguide filledwith a pair of layers made of any two of epsilon-negative (ENG),mu-negative (MNG), double-negative (DNG), and double-positive (DPS)materials in accordance with the invention.

FIG. 1 b illustrates the geometry of a waveguide with a pair ofconcentric layers made of any two of epsilon-negative (ENG), mu-negative(MNG), double-negative (DNG), and double-positive (DPS) materials inaccordance with the invention.

FIG. 2 illustrates a dispersion diagram for TE mode in an ENG-MNGwaveguide illustrating the relationship among normalized d₁, d₂, andnormalized real-valued β_(TE) for two sets of material parameters for apair of ENG-MNG slabs at a given frequency: (a) when ε₁=−2ε₀, μ₁=μ₀,ε₂=3ε₀, μ₂=−2μ₀, and (b) when ε₁=−5ε₀, μ₁=2μ₀, ε₂=2ε₀, μ₂=−μ₀.

FIG. 3 illustrates β_(TM) of the dominant TM mode for thin waveguidesfilled with a pair of DPS-DPS, ENG-MNG, or DPS-DNG slabs, versusγ=d₁/d₂, where in (a) the material parameters are chosen such thatε₁=±2ε₀, μ₁=±μ₀, ε₂=±3ε₀, μ₂=±μ₀ for which |k₂|>|k₁| and in (b) the twoslabs have been interchanged, i.e., slab 1 and 2 in (a) are now slabs 2and 1 in (b), respectively, thus |k₂|<|k₁|.

FIGS. 4 a and 4 b illustrate the lowest admissible values of normalizedd₂ as a function of normalized d₁ and β, for (a) a DPS-DPS waveguide,(b) a DPS-DNG waveguide. ε₁=2, μ₁=1, ε₂=±3, μ₂=±3, TE case, where k₁<k₂.

FIG. 4 c illustrates the multi-branched values of d₂ for the case inFIG. 4 b.

FIGS. 5 a and 5 b illustrate the lowest admissible values of normalizedd₂ as a function of normalized d₁ and β, for (a) a DPS-DPS waveguide,(b) a DPS-DNG waveguide. ε₁=2, μ₁=1, ε₂=±3, μ₂=±3, TM case, where k₁<k₂.

FIG. 6 a illustrates normalized d₂ in terms of normalized d₁ and β, inthe region β>k₂, for TE modes in the waveguide filled with a pair of DPSand DNG slabs with material parameters given as in FIG. 4 b where k₁<k₂.

FIG. 6 b illustrates normalized d₂ in terms of normalized d₁ and β, inthe region β>k₂, for TM modes in the waveguide filled with a pair of DPSand DNG slabs with material parameters given as in FIG. 4 b where k₁<k₂.

FIG. 7 a illustrates normalized d₂ in terms of normalized d₁ and β, inthe region β>k₂, for TE modes in the waveguide filled with a pair of DPSand DNG slabs with material parameters ε₁=3ε₀, μ₁=2μ₀, ε₂=−ε₀, μ₂=−3μ₀,where k₁<k₂.

FIG. 7 b illustrates corresponding quantities for the TM modes in thewaveguide of FIG. 7 a filled with ε₁=−2ε₀, μ₁=−μ₀, ε₂=ε₀, μ₂=3μ₀, wherek₁<k₂.

FIG. 8 illustrates the geometry of the TM wave interaction with twoslabs, one of which can be made of an epsilon-negative (ENG) material,in which real part of permittivity is negative (but real part ofpermeability is positive), and the other made of a mu-negative (MNG)material, in which the real part of permeability can be negative (butreal part of permittivity is positive).

FIG. 9 a illustrates the real and imaginary parts of the normalizedtransverse magnetic field H_(y) as a function of z coordinate when anormally incident TM wave illuminates a sample pair of lossless ENG-MNGslabs.

FIG. 9 b illustrates the real and imaginary parts of the normalizedtransverse magnetic field H_(y) as a function of z coordinate when anormally incident TM wave illuminates a sample pair of lossless MNG-ENGslabs reversed in position from FIG. 9 a.

FIG. 10 a illustrates the real and imaginary parts of the normalizedtransverse magnetic field H_(y) as a function of z coordinate when anormally incident TM wave with a 45° angle of incidence impinges on a“matched pair” of lossless ENG-MNG slabs.

FIG. 10 b illustrates the real and imaginary parts of the normalizedtransverse magnetic field H_(y) as a function of z coordinate when anormally incident TM wave with a 45° angle of incidence impinges on a“matched pair” of lossless MNG-ENG slabs.

FIG. 11 a illustrates the distribution of the real part of the Poyntingvector inside and outside the matched pair of lossless ENG-MNG slabs forthe TM plane wave at

θ_(R = 0)^(TM) = 45^(∘).

FIG. 11 b illustrates the distribution of the real part of the Poyntingvector inside and outside the matched pair of lossless MNG-ENG slabs forthe TM plane wave at

θ_(R = 0)^(TM) = 45^(∘).

FIG. 12 a illustrates a sketch of real and imaginary parts of thenormalized total transverse magnetic field as a function of zcoordinate, when a normally incident TM wave impinges on a “conjugatematched pair” of lossless ENG-MNG slabs.

FIG. 12 b illustrates the distribution of the real and imaginary part ofthe normalized Poynting vector inside and outside of the structure ofFIG. 12 a, where the normalization is with respect to the value of thePoynting vector of the incident wave.

FIG. 13 a illustrates the magnitude of the reflection coefficient for athin slab in response to variations of the angle of incidence for thelossless ENG-MNG pair with several sets of parameters designed to makethe pair transparent at

$\theta_{i} = {\frac{\pi}{4}.}$

FIG. 13 b illustrates the magnitude of the reflection coefficient for athick slab in response to variations of the angle of incidence for thelossless ENG-MNG pair with several sets of parameters designed to makethe pair transparent at

$\theta_{i} = {\frac{\pi}{4}.}$

FIG. 14 a illustrates the magnitude of the reflection coefficient forthe conjugate matched pair of ENG-MNG slabs shown in FIG. 12 a when theloss mechanism is introduced in the permittivity of both slabs as theimaginary part of the permittivity.

FIG. 14 b illustrates the magnitude of the transmission coefficient forthe conjugate matched pair of ENG-MNG slabs shown in FIG. 12 a when theloss mechanism is introduced in the permittivity of both slabs as theimaginary part of the permittivity.

FIG. 15 illustrates the effect of loss on the field distribution in aconjugate matched ENG-MNG pair excited by a normally incident planewave.

FIG. 16 illustrates equivalent transmission line models withcorresponding distributed series and shunt elements, representing the TMwave interaction with (a) a pair of ENG-MNG slabs, and (b) a pair ofDPS-DNG slabs.

FIG. 17 illustrates the correspondence between the cascaded “thin”layers of ENG-MNG slabs and the cascaded “thin” layers of DPS-DNG slabs,using the concept of equivalent transmission lines.

FIG. 18 illustrates a conceptual idea for the image displacement andimage reconstruction with conceptually all the spatial Fouriercomponents preserved using the concept of conjugate matched ENG-MNGpaired slabs.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

The invention will be described in detail below with reference to FIGS.1–18. Those skilled in the art will appreciate that the descriptiongiven herein with respect to those figures is for exemplary purposesonly and is not intended in any way to limit the scope of the invention.All questions regarding the scope of the invention may be resolved byreferring to the appended claims.

Overview

Generally speaking, the present invention relates to devices andcomponents made of adjacent slabs of materials, at least one of which isa material with a negative real permittivity or permeability. As notedabove, methods have been suggested in the prior art for constructingparticulate composite media in which the real part of permittivity canattain negative values in a certain frequency range. Likewise,techniques have been studied for forming passive complex media withnegative real permeability in a given frequency band. To form a mediumwith both real permittivity and permeability being negative, one has tocombine both fabrication techniques, as described in the aforementionedarticles to Shelby et al. and Smith et al. However, constructingwaveguide or scattering components from two slabs of materials, eachwith only one of the material parameters having a negative real part inaccordance with the invention, is not suggested by such prior art. Suchis the subject matter of the present invention.

As for the terminology, for a medium with negative real permittivity(and positive real permeability), the inventors use the term“epsilon-negative (ENG)” medium, while for the medium with negative realpermeability (and positive real permittivity) the inventors choose theterm “mu-negative (MNG)” medium. These are indeed media with“single-negative (SNG)” parameter, as opposed to “double-negative (DNG)”media. In most of the analysis, the media are assumed to be lossless.However, when dissipation is considered, the complex parametersε=ε₀(ε_(r)−jε_(i)) and μ=μ₀(μ_(r)−jμ_(i)) are used where ε_(i) and μ_(i)are non-negative quantities for passive media for the time dependencee^(jωt). The inventors also consider loss to be relatively small, i.e.,ε_(i)<<|ε_(r)| and μ_(i)<<|μ_(r)|. Finally, where the followingdescription discusses only the transverse magnetic (TM) polarizationcase, those skilled in the art will appreciate that similar features andresults are obtained for the transverse electric (TE) case, andvice-versa, by using the duality principle.

As will be explained herein, the present inventors contemplate that theinvention may be used in the microwave and optical (visible) regimes.Those skilled in the art will appreciate that techniques formanufacturing and constructing the devices described herein may bedifferent for the different frequency regimes. It is believed that thoseskilled in the art will apply the appropriate techniques depending uponthe frequency regime of interest.

Conventional double-positive materials such as glass (visible frequencyrange) and quartz (microwave frequency range) are, of course,well-known. DNG materials are not as widely known, particularly in thevisible frequency range. For a description of sample DNG materials inthe microwave regime, see Shelby et al., “Experimental Verification of aNegative Index of Refraction,” Science, Vol. 292, No. 5514, pp. 77–79,2001, and Smith et al, “Composite Medium with Simultaneously NegativePermeability and Permittivity,” Phys. Rev. Lett., Vol. 84, No. 18, pp.4184–4187, 2000. The contents of these references are herebyincorporated by reference in their entireties. Sample ENG materials inthe optical (visible) frequency regime are well-known to those skilledin the art. For example, silver, gold, and plasmonic materials have ENGcharacteristics in the visible frequency range. Artificial materialswith ENG characteristics in the microwave frequency regime are alsoknown to those skilled in the art. For example, see Pendry et al.,“Low-Frequency Plasmons in Thin Wire Structures,” J. of Physics:Condense Matter, Vol. 10, pp. 4785–4809, 1998. The contents of thisreference is hereby incorporated by reference in its entirety. On theother hand, sample MNG materials in the optical (visible) frequencyregime are difficult because if one wants to scale down the “wire andloop” inclusion to the visible wavelength scale, the loss will beprohibitively large. However, artificial materials with MNGcharacteristics in the microwave frequency regime are known to thoseskilled in the art. For example, see Pendry, et al., “Magnetism fromConductors and Enhanced Nonlinear Phenomena,” IEEE Trans. MicrowaveTheory Tech., Vol. 47, No. 11, pp. 2075–2081, November 1999. As will beexplained in more detail below, the present inventors have analyzed indetail the wave interaction with a pair of juxtaposed ENG and MNG slabs,showing interesting properties such as resonances, transparency,anomalous tunneling, and zero reflection. Using appropriate distributedcircuit elements in the transmission-line model for the pair of ENG-MNGlayers, the inventors will explain below the unusual field behavior inthese paired ENG-MNG structures and show that such lossless pairs mayexhibit “interface resonance” phenomena, even though each slab alonedoes not manifest such an effect. Based on the following description,those skilled in the art will appreciate that suitably coupled SNG mediamay offer exciting possibilities in the design of waveguide andscattering devices and components. Since only one of their parametersneeds to be negative in a given frequency range, they may be constructedmore easily than DNG media, for which both parameters should possessnegative real parts in a given band of frequencies. For instance, acollisionless isotropic plasma, whose permittivity may be negative atfrequencies below the plasma frequency may be shown to be an ENG mediumat those frequencies.

The description below also includes an analysis of parallel-plate (FIG.1 a) or concentric (FIG. 1 b) wave-guiding structures filled with pairsof layers made of any two of ENG, MNG, DPS, and DNG materials used toexplore possible unconventional features that depend on the choice ofthe constitutive parameters. In the analysis herein, these materials areassumed to be lossless, homogeneous, and isotropic. It will be shownthat by juxtaposing “conjugate” materials, i.e. materials withcomplementary electromagnetic properties, unusual features may beachieved in such guiding structures. For example, the possibility ofmono-modal propagation in arbitrarily thick parallel-plate waveguidesfilled with a pair of ENG-MNG layers and modes with no cut-off thicknessis highlighted. Although the structures considered herein areparallel-plate or concentric waveguides filled with SNG, DNG, and DPSmedia, those skilled in the art will appreciate that their features mayprovide physical insights into ideas and characteristics for otherwaveguide geometries filled with such metamaterials, with applicationsin the design of waveguide and scattering devices and components.

Also, it is important to note that passive SNG and DNG metamaterials, inwhich permittivity and/or permeability may have negative real parts, areinherently dispersive. Therefore, for passive metamaterials, the realparts of the material parameters may be negative only over a certainband of frequencies, and thus their values may significantly vary withthe frequency. As a result, one should in general take into account thefrequency dependence of such material parameters. However, in order toemphasize the salient features of this type of waveguide withoutresorting to an unnecessary complexity, the frequency of operation ω isfixed for purposes of the description herein and the values ofpermittivity and permeability of SNG, DNG, and DPS materials areconsidered to be given at this given frequency. All other parameters ofthe waveguide such as layer thicknesses and longitudinal wave numbersmay arbitrarily vary.

I. Guided Modes in Waveguides with at Least One SNG Layer

A. Geometry

Consider a parallel-plate waveguide 10, made of two infinitely extendingperfect electrically conducting plates 12, 14 separated by the distanced=d₁+d₂, as shown in FIG. 1 a. This waveguide is filled with a pair ofparallel layers 16, 18 where the first layer is made up of ENG or MNGmaterial and the second layer is made up of DNG material, DPS material,or the SNG material not used in the first layer. A monochromatictime-harmonic variation e^(jωt) is assumed. The two slabs 16, 18 arecharacterized by their thicknesses d₁ and d₂, and constitutiveparameters ε₁, μ₁, and ε₂, μ₂, respectively, which are assumed real, butno assumption on their signs has yet been made. The Cartesian coordinatesystem (x,y,z) is shown in FIGS. 1 a and x is chosen as the direction ofpropagation of guided modes.

Imposing appropriate boundary conditions at y=d₁ and y=−d₂, the electricand magnetic field expressions for the TE^(x) mode may be written as:

$\begin{matrix}{E^{TE} = {\hat{z}E_{0}^{TE}{{\mathbb{e}}^{{- {j\beta}_{TE}}x} \cdot \left\{ \begin{matrix}{{\sin\left( {k_{t2}^{TE}d_{2}} \right)}{\sin\left\lbrack {k_{t1}^{TE}\left( {d_{1} - y} \right)} \right\rbrack}} & {y > 0} \\{{\sin\left( {k_{t1}^{TE}d_{1}} \right)}{\sin\left\lbrack {k_{t2}^{TE}\left( {y + d_{2}} \right)} \right\rbrack}} & {y < 0}\end{matrix} \right.}}} & (1) \\{H^{TE} = {j\;\hat{x}\omega^{- 1}E_{0}^{TE}{{\mathbb{e}}^{{- j}\;\beta_{TE}x} \cdot \left\{ {\begin{matrix}{{{- \mu_{1}^{- 1}}k_{t1}^{TE}{\sin\left( {k_{t2}^{TE}d_{2}} \right)}{\cos\left\lbrack {k_{t1}^{TE}\left( {d_{1} - y} \right)} \right\rbrack}} +} \\{\mu_{2}^{- 1}k_{t2}^{TE}{\sin\left( {k_{t1}^{TE}d_{1}} \right)}{\cos\left\lbrack {k_{t2}^{TE}\left( {y + d_{2}} \right)} \right\rbrack}}\end{matrix},{{- \hat{y}}\omega^{- 1}\beta_{TE}E_{0}^{TE}{{\mathbb{e}}^{{- j}\;\beta_{TE}x} \cdot \left\{ \begin{matrix}{\mu_{1}^{- 1}{\sin\left( {k_{t2}^{TE}d_{2}} \right)}{\sin\left\lbrack {k_{t1}^{TE}\left( {d_{1} - y} \right)} \right\rbrack}} & {y > 0} \\{\mu_{2}^{- 1}{\sin\left( {k_{t1}^{TE}d_{1}} \right)}{\sin\left\lbrack {k_{t2}^{TE}\left( {y + d_{2}} \right)} \right\rbrack}} & {y < 0}\end{matrix} \right.}}} \right.}}} & (2)\end{matrix}$where E₀ ^(TE) is the mode amplitude, determined by the excitation, andk_(ti) ^(TE)=√{square root over (k_(i) ²−β_(TE) ²)} with k_(i)²=ω²μ_(i)ε_(i) for i=1, 2. The corresponding expressions for the TM^(x)modes may be easily obtained (not shown here). For ENG and MNG slabs,where one of the material parameters is negative, k_(i) ²<0, and forpropagating modes with real β, the transverse wave number k_(ti) isalways imaginary. However, for DPS and DNG slabs, k_(i) ²>0 and thetransverse wave number k_(ti) may be real or imaginary, depending on thevalue of β. The field expressions in (1) and (2) and the correspondingexpressions for the TM case remain valid for any of these cases. (Forsimplicity, the superscript ‘x’ in TE^(x) and TM^(x) will be droppedhereafter.) By applying the boundary conditions for the tangentialcomponents of the electric and magnetic fields at the interface y=0, onefinds the following two dispersion relations for the TE and TM modes,respectively:

$\begin{matrix}{{\frac{\mu_{1}}{k_{t1}^{TE}}{\tan\left( {k_{t1}^{TE}d_{1}} \right)}} = {{- \frac{\mu_{2}}{k_{t2}^{TE}}}{\tan\left( {k_{t2}^{TE}d_{2}} \right)}}} & (3) \\{{\frac{ɛ_{1}}{k_{t1}^{TM}}{\cot\left( {k_{t1}^{TM}d_{1}} \right)}} = {{- \frac{ɛ_{2}}{k_{t2}^{TM}}}{{\cot\left( {k_{t2}^{TM}d_{2}} \right)}.}}} & (4)\end{matrix}$

Depending on the choice of material parameters, the above dispersionrelations reveal interesting characteristics for the guided modespresent in this waveguide. Some of the features of propagating guidedmodes for various pairs of ENG, MNG, DPS, and/or DNG slabs filling thisguiding structure in accordance with the invention will be explainedbelow.

B. Pairs of SNG Slabs: ENG-MNG, ENG-ENG, and MNG-MNG Pairs

For a pair of ENG and MNG slabs, ε_(1<0, μ) ₁>0, ε₂>0, and μ₂<0, andthus k₁ ²=ω²μ₁ε₁<0 and k₂ ²=ω²μ₂ε₂<0. For propagating modes, β_(TE) andβ_(TM) should be real-valued quantities, and thus k_(ti)=j√{square rootover (|k_(i)|²+β²)} is purely imaginary for i=1, 2. The dispersionrelations (3) and (4) above for the TE and TM modes can then berewritten, respectively, as:

$\begin{matrix}{{\frac{\mu_{1}}{\sqrt{\left| k_{1} \middle| {}_{2}{+ \beta_{TE}^{2}} \right.}}{\tanh\left( {\sqrt{\left| k_{1} \middle| {}_{2}{+ \beta_{TE}^{2}} \right.}d_{1}} \right)}} = {{- \frac{\mu_{2}}{\sqrt{\left| k_{2} \middle| {}_{2}{+ \beta_{TE}^{2}} \right.}}}{\tanh\left( {\sqrt{\left| k_{2} \middle| {}_{2}{+ \beta_{TE}^{2}} \right.}d_{2}} \right)}}} & (5) \\{{\frac{ɛ_{1}}{\sqrt{\left| k_{1} \middle| {}_{2}{+ \beta_{TM}^{2}} \right.}}{\coth\left( {\sqrt{\left| k_{1} \middle| {}_{2}{+ \beta_{TM}^{2}} \right.}d_{1}} \right)}} = {{- \frac{ɛ_{2}}{\sqrt{\left| k_{2} \middle| {}_{2}{+ \beta_{TM}^{2}} \right.}}}{{\coth\left( {\sqrt{\left| k_{2} \middle| {}_{2}{+ \beta_{TM}^{2}} \right.}d_{2}} \right)}.}}} & (6)\end{matrix}$

For the case where one slab is a lossless ENG and the other is alossless MNG, the goal is to find the conditions under which one mayobtain real-valued solutions for the longitudinal wave number β. Due tothe monotonic behavior and the asymptotic limit of the hyperbolictangent and cotangent functions with real argument in (5) and (6), oneexpects to observe interesting dispersion characteristics. First,because (5) and (6) are indeed valid for any pair of SNG materials, ifμ₁ and μ₂ have the same sign, (5) may not have any real-valued solutionfor β_(TE). Likewise, if ε₁ and ε₂ have the same sign, there may not beany TM mode with real β_(TM) from (6). This is not surprising, since forthe case of the ENG-ENG pair or the MNG-MNG pair, where permittivitiesand permeabilities have the same signs, i.e., ε₁ε₂>0 and μ₁μ₂>0, thewave number in the bulk paired materials is always imaginary. However,when one has a so-called “conjugate” pair, i.e., a pair of ENG and MNGslabs in this waveguide, both sides of (5) and (6) have the same sign,and thus it is possible to have real-valued solutions for β_(TE) andβ_(TM).

In order to gain some physical insights into the possible real-valuedsolutions of (5) and (6), one may assume that for a given set ofmaterial parameters for the two slabs, the thickness of the ENG slab d₁is known and fixed, and one may find the MNG slab thickness d₂ such thatβ attains a specific real value. This can be found by rearranging (5)and (6) as:

$\begin{matrix}{d_{2}^{TE} = \frac{\tanh^{- 1}\left\lbrack {\frac{\left| \mu_{1} \middle| \sqrt{\left| k_{2} \middle| {}_{2}{+ \beta_{TE}^{2}} \right.} \right.}{\left| \mu_{2} \middle| \sqrt{\left| k_{1} \middle| {}_{2}{+ \beta_{TE}^{2}} \right.} \right.}{\tanh\left( {\sqrt{\left| k_{1} \middle| {}_{2}{+ \beta_{TE}^{2}} \right.}d_{1}} \right)}} \right\rbrack}{\sqrt{\left| k_{2} \middle| {}_{2}{+ \beta_{TE}^{2}} \right.}}} & (7) \\{{d_{2}^{TM} = \frac{\tanh^{- 1}\left\lbrack {\frac{\left| ɛ_{2} \middle| \sqrt{\left| k_{1} \middle| {}_{2}{+ \beta_{TM}^{2}} \right.} \right.}{\left| ɛ_{1} \middle| \sqrt{\left| k_{2} \middle| {}_{2}{+ \beta_{TM}^{2}} \right.} \right.}{\tanh\left( {\sqrt{\left| k_{1} \middle| {}_{2}{+ \beta_{TM}^{2}} \right.}d_{1}} \right)}} \right\rbrack}{\sqrt{\left| k_{2} \middle| {}_{2}{+ \beta_{TM}^{2}} \right.}}},} & (8)\end{matrix}$A physical solution for d₂ exists only if the argument of the inversehyperbolic tangent function in the above equations is between zero andunity, which suggests that not for every arbitrary pair of d₁ and β onewill be able to find a solution for d₂. But when this condition isfulfilled, the solution for d₂ is unique due to the monotonic(non-periodic) behavior of the hyperbolic tangent function.Specifically, for a given set of parameters for the pair of ENG and MNGslabs and a fixed d₁, there may only be one value for d₂ ^(TE) when agiven real value for β_(TE) is desired, and similarly there may only beone value for d₂ ^(TM) when β_(TM) is given. These features are incontrast with those of propagating modes in a conventional waveguidefilled with a pair of DPS-DPS slabs, where one has multiple solutionsfor d₂ due to the periodic behavior of the fields in the transversesection. Furthermore, in the ENG-MNG waveguide the field components varyas hyperbolic sinusoidal functions in the transverse plane, and they aremostly concentrated around the ENG-MNG interface. As will be shownbelow, similar field distributions and concentration near the interfacemay be observed in DPS-DNG waveguides when β>max(|k₁|,|k₂|), as alsoobserved by Nefedov and Tretyakov in an article entitled “Waveguidecontaining a backward-wave slab,” e-print in arXiv:cond-mat/0211185 v1,at http://arxiv.org/pdf/cond-mat/0211185, 10 Nov. 2002. As will be notedbelow, there are other interesting properties in the DPS-DNG waveguidesthat may resemble those of the ENG-MNG waveguides discussed here.

The conditions for having the arguments of the inverse hyperbolictangent functions in (7) and (8) less than unity may be explicitly givenas:

$\begin{matrix}{{\tanh\left( {\sqrt{\left. {\beta_{TE}^{2} +} \middle| k_{1} \right|^{2}}d_{1}^{TE}} \right)} < \frac{\left| \mu_{2} \middle| \sqrt{\left. {\beta_{TE}^{2} +} \middle| k_{1} \right|^{2}} \right.}{\left| \mu_{1} \middle| \sqrt{\left. {\beta_{TE}^{2} +} \middle| k_{2} \right|^{2}} \right.}} & (9) \\{{\tanh\left( {\sqrt{\left. {\beta_{TM}^{2} +} \middle| k_{1} \right|^{2}}d_{1}^{TM}} \right)} < {\frac{\left| ɛ_{1} \middle| \sqrt{\left. {\beta_{TM}^{2} +} \middle| k_{2} \right|^{2}} \right.}{\left| ɛ_{2} \middle| \sqrt{\left. {\beta_{TM}^{2} +} \middle| k_{1} \right|^{2}} \right.}.}} & (10)\end{matrix}$

If the terms on the right-hand side of (9) and (10) are greater thanunity, then any values of d₁ ^(TE) and d₁ ^(TM), no matter how large orsmall, may satisfy these inequalities. However, if the right-hand termsare less than unity, then only certain limited ranges of d₁ ^(TE) and d₁^(TM) may fulfill (9) and (10). It is interesting to note that, due tothe symmetric nature of the dispersion relations, when d₁ is limited toa finite range, d₂ will have a unique solution between zero and infinityand, vice versa, if d₂ is limited, then d₁ may find a unique solution inthat infinite range. In other words, for any given β (for TE or TM) onlyone of the two corresponding thicknesses d₁ and d₂ may be confined to afinite range of variation.

One special case, namely when the right-hand side of (9) or (10) becomesunity, deserves particular attention, since in such a case neither d₁nor d₂ is limited to a finite range of variation. The values of β thatmay provide this special condition is explicitly given by:

$\begin{matrix}{\beta_{sw}^{TE} = {{\pm \omega}\sqrt{\frac{{ɛ_{1}/\mu_{1}} - {ɛ_{2}/\mu_{2}}}{\mu_{1}^{- 2} - \mu_{2}^{- 2}}}}} & (11) \\{{\beta_{sw}^{TM} = {{\pm \omega}\sqrt{\frac{{\mu_{1}/ɛ_{1}} - {\mu_{2}/ɛ_{2}}}{ɛ_{1}^{- 2} - ɛ_{2}^{- 2}}}}},} & (12)\end{matrix}$where ε₁μ₁<0 and ε₂μ₂<0 for the ENG and MNG slabs. When, with properchoices of material parameters, β_(sw) ^(TE) or β_(sw) ^(TM) arereal-valued quantities, they represent the wave numbers for the TE or TMsurface wave that may exist along the interface of the two semi-infinitelossless ENG and MNG media. These relations are formally similar to thewave number expressions for the surface waves supported at the interfaceof the DPS and DNG half spaces described by the present inventors in“Radiation from a traveling-wave current sheet at the interface betweena conventional material and a material with negative permittivity andpermeability,” Microwave and Opt. Tech. Lett., Vol. 35, No. 6, pp.460–463, Dec. 20, 2002, and by Lindell et al. in “BW media—media withnegative parameters, capable of supporting backward waves,” Microwaveand Opt. Tech. Lett., Vol. 31, No. 2, pp. 129–133, 2001. (Strictlyspeaking, in the latter case the expressions (11) and (12) represent thewave numbers of surface waves only if these values of β_(sw) are realand greater than both k₁ and k₂ of the DPS and DNG media. In the ENG-MNGcase, however, since k₁ and k₂ are both imaginary, as long as either ofthese β_(sw) is real, a surface wave may propagate.) From (11) and (12),the conditions for a given interface between ENG and MNG media tosupport a TE or TM surface wave can be expressed as:

$\begin{matrix}{{TE}\text{:}\left\{ {\begin{matrix}\left| \mu_{1} \middle| {< \left| \mu_{2} \right|} \right. \\\left| \eta_{1} \middle| {> \left| \eta_{2} \right|} \right.\end{matrix}\mspace{14mu}{or}\mspace{14mu}\left\{ {\begin{matrix}\left| \mu_{1} \middle| {> \left| \mu_{2} \right|} \right. \\\left| \eta_{1} \middle| {< \left| \eta_{2} \right|} \right.\end{matrix},{{TM}\text{:}\left\{ {\begin{matrix}\left| ɛ_{1} \middle| {< \left| ɛ_{2} \right|} \right. \\\left| \eta_{1} \middle| {< \left| \eta_{2} \right|} \right.\end{matrix}\mspace{14mu}{or}\mspace{14mu}\left\{ \begin{matrix}\left| ɛ_{1} \middle| {> \left| ɛ_{2} \right|} \right. \\\left| \eta_{1} \middle| {> \left| \eta_{2} \right|} \right.\end{matrix} \right.} \right.}} \right.} \right.} & (13)\end{matrix}$

-   where η_(i) (i=1, 2) denotes the intrinsic impedance of the medium,    which is an imaginary quantity for ENG and MNG materials.    Conditions (13) imply that no interface may support both TE and TM    surface waves, i.e., either a TE or a TM surface wave may be    supported, but not both.

FIG. 2 illustrates the TE dispersion diagram of such an ENG-MNGwaveguide for two different sets of material parameters. In particular,FIG. 2 illustrates the relationship among normalized d₁, d₂, andnormalized real-valued β_(TE), as described in (7), for two sets ofmaterial parameters for a pair of ENG-MNG slabs at a given frequency:FIG. 2 a when ε₁=−2ε₀, μ₁=μ₀, ε₂=3ε₀, μ₂=−2μ₀, and FIG. 2 b whenε₁=−5ε₀, μ₁=2μ₀, ε₂=2ε₀, μ₂=−μ₀. The set of material parameters chosenin FIG. 2 a does not allow a TE surface wave at the ENG-MNG interface,while the set chosen in FIG. 2 b does. The value of d′₂ is given by (18)below. One striking feature of these diagrams, as already mentioned, isthe single-valuedness of d₂ ^(TE) for a given set of d₁ and β_(TE). Inorder to understand these figures better, some special limits will beexplained.

1. Thick Waveguides

At one extreme, assume |k₁|d₁ and |k₂|d₂ to be large. Equations (5) and(6) may then be simplified as:

$\begin{matrix}{\frac{\left| \mu_{1} \right|}{\sqrt{\left| k_{1} \middle| {}_{2}{+ \beta_{TE}^{2}} \right.}} \simeq \frac{\left| \mu_{2} \right|}{\sqrt{\left| k_{2} \middle| {}_{2}{+ \beta_{TE}^{2}} \right.}}} & (14) \\{{\frac{\left| ɛ_{1} \right|}{\sqrt{\left| k_{1} \middle| {}_{2}{+ \beta_{TM}^{2}} \right.}} \simeq \frac{\left| ɛ_{2} \right|}{\sqrt{\left| k_{2} \middle| {}_{2}{+ \beta_{TM}^{2}} \right.}}},} & (15)\end{matrix}$

-   which are independent of the slab thicknesses. Solving these    equations for β_(TE) and β_(TM), one obtains β_(sw) ^(TE) and β_(sw)    ^(TE) as given in (11) and (12), respectively. This is physically    justified, since for |k₁|d₁>>1 and |k₂|d₂>>1, the waveguide walls    are far apart from the interface between the two slabs, and because    the fields are concentrated around this interface, the ENG-MNG pair    effectively behaves as two semi-infinite regions. Therefore,    provided that a mode is supported by such a thick structure (and    this depends whether the interface may support a surface wave), this    mode should resemble such a surface wave. This may be seen in FIG. 2    b, where in the limit of very large |k₁|d₁ and |k₂|d₂, the wave    number β_(TE) approaches the value given in (11). It is worth noting    that the curved line, beyond which the dispersion diagram in FIG. 2    b “stops” and along which it diverges, is defined by the boundary of    the region satisfying (9). Beyond this region, for a given pair of    d₁ and β, no real solution for d₂ may be obtained from (7).    Analogous features may be observed for the TM case.    2. Thin Waveguides

If the thicknesses |k₁|d₁ and |k₂|d₂ are assumed to be very small, (3)and (4) may be approximated by:

$\begin{matrix}{\gamma \simeq {- \frac{\mu_{2}}{\mu_{1}}}} & (16) \\{{\beta_{TM} \simeq {{\pm \omega}\sqrt{\frac{{\mu_{1}\gamma} + \mu_{2}}{{\gamma/ɛ_{1}} + {1/ɛ_{2}}}}}},} & (17)\end{matrix}$

-   where γ is a shorthand for d₁/d₂ and should always be a positive    quantity. These approximate expressions are valid for thin    waveguides loaded with any pair of slabs, since they have been    obtained directly from (3) and (4). This point is physically    justified considering the fact that in thin waveguides the    transverse behavior of the field, which determines the possibility    of a mode to propagate, is similar for DPS, DNG and SNG materials,    since the hyperbolic and trigonometric sinusoidal functions have    somewhat similar behavior in the limit of small arguments.

For a thin waveguide filled with a pair of DPS-DPS layers (and similarlywith a pair of ENG-ENG, DPS-ENG, MNG-MNG, DNG-MNG, or DNG-DNG layers),(16) may never be satisfied, because for these pairs μ₂/μ₁>0 and thus noTE mode may propagate in such a thin waveguide, as expected. On theother hand, (17) will provide the approximate value for ATM of thedominant TM mode, if β_(TM) turns out to be a real quantity for a givenset of γ and material parameters. β_(TM) depends on the ratio of layerthicknesses, not on the total thickness. Therefore, this TM mode has nocut-off thickness, i.e., there is not a thickness below which the TMmode may not propagate. For a DPS-DPS or DNG-DNG thin waveguide, this TMmode exists for any ratio γ, and its β_(TM) is sandwiched between k₁ andk₂, which are effectively the two limits of (17) for γ→∞ and γ→0,respectively. This implies that the TM field distribution in thetransverse section of a DPS-DPS or DNG-DNG thin waveguide has to beexpressed using the exponential functions in one of the two slabs (inthe one with smaller wave number) and the sinusoidal functions in theother slab. The allowable ranges of variation of β_(TM) in (17) in termsof γ are shown in FIG. 3 for various pairs of slabs.

FIG. 3 illustrates β_(TM) of the dominant TM mode for thin waveguidesfilled with a pair of DPS-DPS, ENG-MNG, or DPS-DNG slabs, versusγ=d₁/d₂. In FIG. 3 a, the material parameters are chosen such thatε₁=±2ε₀, μ₁=μ₀, ε₂=±3ε₀, μ₂=±μ₀ for which |k₂|>|k₁|; in FIG. 3 b, thetwo slabs have been interchanged, i.e., slab 1 and 2 in FIG. 3 a are nowslabs 2 and 1 in FIG. 3 b, respectively, thus |k₂|<|k₁|. FIG. 3 b isconcerned only with the positive real solutions for β_(TM), but itsnegative real solutions are simply obtained by flipping its sign. Asshown in FIG. 3, the ENG-MNG pair behaves differently: the existence ofa no-cut-off dominant TM mode is restricted to the waveguides with γ inthe range between |ε₁|/|ε₂| and |μ₂|/|μ₁|. However, its wave numberβ_(TM) is not restricted to any interval, i.e., an ENG-MNG waveguide mayhave a dominant no-cut-off TM mode with β_(TM) ranging from zero toinfinity.

In the TE case, a thin waveguide with a pair of ENG-MNG slabs (or also apair of DNG-ENG, DPS-MNG, or DPS-DNG slabs), has μ₂/μ₁<0, and thus (16)may be satisfied for a certain value of γ. Equation (16) seems to beeffectively independent of β_(TE). However, in such a limit, the wavenumber β_(TE) of the guided mode may essentially attain any real value,as can be seen in FIGS. 2 a and 2 b around the region where |k₁|d₁→0 and|k₂|d₂→0. In such a limit, no matter how thin these layers are (as longas they satisfy (16)), one (and only one) propagating mode may exist. Inother words, this waveguide does not have a cut-off thickness for the TEmodes. This feature represents a generalization of the analysis for theDPS-DNG thin cavity previously described by the present inventors. Forthin layers of ENG and MNG slabs, when d₁ and d₂ are selected to satisfyrelation (16), the particular solution for β_(TE), which is unique, maybe obtained by solving (5).

Another interesting feature to note in FIG. 2 is the relationshipbetween d₁ and d₂ at β=0, which is the case of a 1-dimensional cavityfilled with the ENG-MNG pair. When d₁ is chosen to be large, d₂ willapproach the finite limit given by:

$\begin{matrix}{{d_{2} = {\frac{1}{\left| k_{2} \right|}{\tanh^{- 1}\left( \sqrt{\frac{\mu_{1}ɛ_{2}}{\mu_{2}ɛ_{1}}} \right)}}},} & (18)\end{matrix}$

-   provided that

$\sqrt{\frac{\mu_{1}ɛ_{2}}{\mu_{2}ɛ_{1}}} < 1$(which is the case for the parameters used in FIG. 2). This relation isobviously the same for both polarizations, since their behaviorscoincide when β→0. Depending on the choice of the material parameters,for a fixed d₁, when one increases β from zero the thickness d₂ usuallydoes not show local minima or maxima, but it monotonically decreases orincreases, as can be seen in FIGS. 2 a and 2 b, respectively. Thisfeature implies the existence of a single mode in such an ENG-MNGwaveguide for a given set of d₁ and d₂. It is important to point outthat such a mono-modal characteristic is effectively independent of thewaveguide total thickness. For instance, from FIG. 2 a, one can see thatfor a given β_(TE) and a specific allowable d₂, thickness d₁ may bechosen very large, resulting in a thick waveguide. But still only onesingle mode is propagating in such a thick waveguide. This feature, notpresent in a conventional waveguide, may be potentially employed forpossible applications in the design of mono-modal waveguides with alarge aperture. It can be shown that the mono-modality property ispresent in any ENG-MNG waveguide, whose interface (between the ENG andMNG media) may support a surface wave (and therefore satisfies (13)),and in most (although not all) of the other ENG-MNG waveguides whoseinterface may not support a surface wave.C. Pairs of DNG and DPS Slabs: DPS-DNG, DPS-DPS, and DNG-DNG Pairs

This section will highlight some of the features of guided modes inknown parallel-plate waveguides filled with various pairs of DPS and DNGlayers, and then compare and contrast these features of such knownwaveguides with those of the waveguides with SNG pairs, as discussedabove, in accordance with the invention. In this case, for lossless DPSand DNG slabs, εμ>0, and thus k_(i) ²=ω²μ_(i)ε_(i)>0 for i=1, 2.

1. Thin Waveguides

In many regards, various features of the thin DPS-DNG waveguide resemblethose of the thin ENG-MNG waveguide. In fact, (16) and (17) againprovide the approximate dispersion relations for the TE and TM modes inthe DPS-DNG case. The TE polarization in this case is thoroughlyequivalent with that of the thin ENG-MNG case. However, the TM mode heredeserves further discussion. As may be seen from FIG. 3, the range ofvariation of β_(TM) in the thin DPS-DNG waveguide differs from the onesin the thin ENG and in the standard DPS-DPS waveguides. Here β_(TM) mayattain values only outside the interval between |k₁| and |k₂|(effectively “complementary” to the standard DPS-DPS case where β_(TM)is in this interval), and γ should also be outside the range between−μ₂/μ₁ and −ε₁/ε₂ The fact that thin waveguides loaded with “conjugate”pairs of metamaterials (e.g., DPS-DNG or ENG-MNG) may supportnon-limited β_(TM), may offer interesting possibilities in designingvery thin resonant cavities, as already proposed by the presentinventors in the DPS-DNG case, for which β=0 when γ=−μ₂/μ₁, or for verythin waveguides having guided modes with high β. A similar observationregarding the possibility of β_(TM) to be very large has also been madeby Nefedov et al.

2. Waveguides with Arbitrary Thickness

The wave numbers k₁ and k₂ assume real values in lossless DPS and DNGlayers. Therefore, one may consider three distinct intervals for thelongitudinal wave number β:

I. β<min(|k₁|,|k₂|)

In this interval, the transverse wave numbers k_(i1) and k_(i2) are bothreal. Equations (3) and (4) may then be rearranged as follows to expressthe value of d₂ in terms of other parameters:

$\begin{matrix}{d_{2}^{TE} = \frac{{\tan^{- 1}\left\lbrack {{- \frac{\mu_{1}\sqrt{k_{2}^{2} - \beta^{2}}}{\mu_{2}\sqrt{k_{1}^{2} - \beta^{2}}}}{\tan\left( {\sqrt{k_{1}^{2} - \beta^{2}}d_{1}} \right)}} \right\rbrack} + {m\;\pi}}{\sqrt{k_{2}^{2} - \beta^{2}}}} & (19) \\{{d_{2}^{TM} = \frac{{\tan^{- 1}\left\lbrack {{- \frac{ɛ_{2}\sqrt{k_{1}^{2} - \beta^{2}}}{ɛ_{1}\sqrt{k_{2}^{2} - \beta^{2}}}}{\tan\left( {\sqrt{k_{1}^{2} - \beta^{2}}d_{1}} \right)}} \right\rbrack} + {m\mspace{2mu}\pi}}{\sqrt{k_{2}^{2} - \beta^{2}}}},} & (20)\end{matrix}$

-   where m is an integer. From these equations, it may be noted that    for a given set of material parameters, when β and d₁ are fixed, the    thickness d₂ for which a mode is supported has infinite solutions in    DPS-DPS, DPS-DNG, and DNG-DNG waveguides.    II. min(|k₁|,|k₂|)<β<max(|k₁|,|k₂|)

For |k₁|<β<|k₂|, (19) and (20) are modified as:

$\begin{matrix}{d_{2}^{TE} = \frac{{\tan^{- 1}\left\lbrack {{- \frac{\mu_{1}\sqrt{k_{2}^{2} - \beta^{2}}}{\mu_{2}\sqrt{\beta^{2} - k_{1}^{2}}}}{\tanh\left( {\sqrt{\beta^{2} - k_{1}^{2}}d_{1}} \right)}} \right\rbrack} + {m\;\pi}}{\sqrt{k_{2}^{2} - \beta^{2}}}} & (21) \\{d_{2}^{TM} = {\frac{{\tan^{- 1}\left\lbrack {{- \frac{ɛ_{2}\sqrt{\beta^{2} - k_{1}^{2}}}{ɛ_{1}\sqrt{k_{2}^{2} - \beta^{2}}}}{\tanh\left( {\sqrt{\beta^{2} - k_{1}^{2}}d_{1}} \right)}} \right\rbrack} + {m\mspace{2mu}\pi}}{\sqrt{k_{2}^{2} - \beta^{2}}}.}} & (22)\end{matrix}$

(If |k₂|<β<|k₁| similar expressions may be obtained, but in this case d₂^(TM) and d₂ ^(TE) will be expressed in terms of inverse hyperbolictangent functions, and thus will be single-valued). It should bementioned that the minus sign in the argument of the inverse tangentfunctions has disappeared in (22). This is related to the fact that inthe thin waveguide approximation discussed earlier, a DPS-DPS waveguidemay support a dominant no-cut-off TM mode in this range (k₁<β_(TM)<k₂),while a thin DPS-DNG waveguide may not (see FIG. 3). In fact, if onetakes the limit of (22) for d₁ very small,

$d_{2}^{TM} \cong {{\frac{ɛ_{2}\left( {\beta^{2} - k_{1}^{2}} \right)}{ɛ_{1}\left( {k_{2}^{2} - \beta^{2}} \right)}d_{1}} + \frac{m\;\pi}{\sqrt{k_{2}^{2} - \beta^{2}}}}$and, when ε₂/ε₁<0 (DPS-DNG case), the first admissible value for m isunity, which implies that d₂ ^(TM) cannot be arbitrarily small, and thusnot allowing a dominant TM mode with β_(TM) in the range (k₁<β_(TM)<k₂)for a thin DPS-DNG waveguide.

FIGS. 4 a and 4 b illustrate plots of the lowest admissible values ofnormalized d₂ from (19) and (21), for a given set of materialparameters, as a function of normalized d₁ and β_(TE) for the TE casefor: (a) a DPS-DPS waveguide, and (b) a DPS-DNG waveguide. ε₁=2, μ₁=1,ε₂=±3, μ₂=±3 in the TE case. The parameters have been chosen so that|k₁|<|k₂|. (Analogous corresponding results may be obtained if|k₂|<├k₁|). FIG. 4 c shows the multi-branched values of d₂ for the casein FIG. 4 b. FIG. 4 b is essentially taken from FIG. 4 c in that onlythe lowest values of d₂ are shown in FIG. 4 b.

In connection with FIG. 4, it is important to reiterate that d₂ in(19)–(22) are multi-valued for any given set of parameters, and thus theplots of d₂ should be multi-branched. (An example of such multi-branchedplots is shown in FIG. 4 c, which is for the DPS-DNG waveguide of FIG. 4b). However, FIGS. 4 a and 4 b show only the lowest positive values ofd₂ for given parameters. The discontinuities observed are due to thejumps from one branch of d₂ to another, in order to attain the lowestvalue for d₂; however each branch of d₂ by itself is indeed continuous,as evident from FIG. 4 c. One can see from FIG. 4 a, for standardDPS-DPS waveguides, that as d₁→0 the lowest admissible value of d₂ tosupport a mode approaches a non-zero value of π/k_(i2), and that thisvalue expectedly becomes infinitely large as β→k₂, since k_(i2)→0. Whend₂→0, on the other hand, the value of d₁ satisfies the relation

${d_{1} = {\frac{p\;\pi}{k_{t1}} = \frac{p\;\pi}{\sqrt{k_{1}^{2} - \beta^{2}}}}},$where p is a positive integer. This locus can be clearly seen in FIG. 4.Thus, in a DPS-DPS waveguide, obviously no TE mode may exist if

${{d_{1} + d_{2}} < {\min\left( {\frac{\pi}{k_{t1}},\frac{\pi}{k_{t2}}} \right)}},$which implies that, as is well known, there is a constraint on theminimum total thickness of any DPS-DPS waveguide in order to have a TEmode.

For DPS-DNG waveguides, however, the situation differs markedly. Becausea DNG slab may act as a phase compensator, effectively canceling thephase delay of a DPS slab, a DPS-DNG waveguide may have a TE mode asboth d₁ and d₂ approach zero, as can be seen in FIGS. 4 b and 4 c. Theconstraint is on the ratio of the transverse phase delays in the twoslabs, not on their sum. For the thin waveguide approximation, in asimilar way, this constraint is manifested as the ratio γ=d₁/d₂, not thesum d₁+d₂, and it is described in (16) and (17). This is reflected inthe slope of the curve in FIG. 4 b in the vicinity of d₁→0 and d₂→0. Thelocus for d₂₌₀ has the same expression as in the DPS-DPS case, but herep in the expression d₁=pπ/√{square root over (k₁ ²−β²)} may also bezero. However, in FIG. 4 b, it may be noted that as β→k₂−0, the lowestpositive value of d₂ does not become infinitely large. This can beunderstood by evaluating the limit of (21) for β→k₂−0 as follows:

$\begin{matrix}{{\lim\limits_{\beta->{k_{2} - 0}}d_{2}^{TE}} = {{{- \frac{\mu_{1}}{\mu_{2}}}\frac{\tanh\left( {\sqrt{\beta^{2} - k_{1}^{2}}d_{1}} \right)}{\sqrt{\beta^{2} - k_{1}^{2}}}} + {\frac{m\;\pi}{\sqrt{k_{2}^{2} - \beta^{2}}}.}}} & (23)\end{matrix}$

For a DPS-DNG waveguide, where −μ₁/μ₂ is a positive quantity, thesmallest positive value for d₂ ^(TE) from (23) is obtained when m=0.This is indeed what is shown in FIG. 4 b in the neighborhood of β→k₂−0.Upper branches of d₂ ^(TE) (not shown in FIG. 4 b, but shown in FIG. 4c), for which m≧1, approach +∞ as β→k₂−0 due to the term

$\frac{m\;\pi}{\sqrt{k_{2}^{2} - \beta^{2}}}.$In the DPS-DPS case, the term −μ₁/μ₂ is a negative quantity, andtherefore the lowest value of d₂ ^(TE) in (23) is obtained when m=1,which causes d₂ ^(TE)→∞ as β→k₂−ε for every branch.

Another interesting observation to be made with respect to FIG. 4 is therelationship between d₁ and d₂ when β=0. This “β=0 cut” in the figuresrepresents the dispersion characteristics of a cavity resonator filledwith a pair of DPS-DPS layers (FIG. 4 a) and DPS-DNG layers (FIG. 4 b).The possibility of having a thin sub-wavelength cavity resonator with apair of thin DPS and DNG slabs may be seen in FIG. 4 b for β=0, as d₁→0and d₂→0.

FIG. 5 presents the corresponding plots of FIG. 4 for the TM case.Similar to FIG. 4, in these plots the lowest admissible values ofnormalized d₂ from (20) and (22) are shown as a function of normalizedd₁ and β_(TM). The material parameters are the same as in FIG. 4, andare chosen such that |k₁|<|k₂|. Some of the main differences between theTE and TM modes can be observed by comparing the two figures. Forinstance, the expected presence of a no-cut-off dominant TM mode in thethin DPS-DPS waveguide may be observed in FIG. 5 a for d₁→0, d₂→0, andk₁<β_(TM)<k₂, and the absence of this mode in the thin DPS-DNG case inFIG. 5 b in the range k₁<β_(TM)<k₂. The variation of γ=d₁/d₂ withβ_(TM), in the limit of d₁→0, d₂→0, in the range k₁<β_(TM)<k₂ for theDPS-DPS case may also be seen in FIG. 5 a, and in the range β_(TM)<k₁for the DPS-DNG case in FIG. 5 b, all according to (17) and FIG. 3. Inparticular, it is observed that in thin DPS-DPS waveguides the ratiod₁/d₂ attains all real values from 0 to +∞ as β_(TM) varies in theadmissible interval between k₁ and k₂ (FIG. 5 a), whereas in the thinDPS-DNG case when k₁<k₂ (FIG. 5 b) the ratio d₁/d₂ varies from −μ₂/μ₁(when β_(TM)=0) to ∞ (when β_(TM)=k₁), and when k₁>k₂ (not shown here)the ratio d₁/d₂ goes from zero (when β→k₂) to −μ₂/μ₁ (when β_(TM)=0),all consistent with (17) and FIG. 3.

III. max(|k₁|,|k₂|)<β

In a conventional DPS-DPS waveguide, one may not have a guided mode withreal-valued β in this range. The same is true for a DNG-DNG waveguide.However, if one of the slabs is made of a lossless material with oneand/or both of its permittivity and permeability negative, it will thenbe possible to have a TE and/or a TM guided mode, as shown below. Forthis range of β, the dispersion relations in (3) and (4) may berewritten as follows:

$\begin{matrix}{d_{2}^{TE} = \frac{\tanh^{- 1}\left\lbrack {{- \frac{\mu_{1}\sqrt{\beta^{2} - k_{2}^{2}}}{\mu_{2}\sqrt{\beta^{2} - k_{1}^{2}}}}{\tanh\left( {\sqrt{\beta^{2} - k_{1}^{2}}d_{1}} \right)}} \right\rbrack}{\sqrt{\beta^{2} - k_{2}^{2}}}} & (24) \\{d_{2}^{TM} = {\frac{\tanh^{- 1}\left\lbrack {{- \frac{ɛ_{2}\sqrt{\beta^{2} - k_{1}^{2}}}{ɛ_{1}\sqrt{\beta^{2} - k_{2}^{2}}}}{\tanh\left( {\sqrt{\beta^{2} - k_{1}^{2}}d_{1}} \right)}} \right\rbrack}{\sqrt{\beta^{2} - k_{2}^{2}}}.}} & (25)\end{matrix}$

These equations are analogous to (7) and (8) derived for the ENG-MNGwaveguide, and they exhibit similar features. (The next section willgive further insights into this analogy.) In both cases, β²>k_(i) ² withi=1, 2 and the field distributions in the transverse plane are in termsof hyperbolic sinusoidal functions, and in both cases d₂ issingle-valued. The characteristics of the surface waves are also similarin both waveguides, and the formal expressions for β_(sw) ^(TE) andβ_(sw) ^(TM) in (11) and (12) are still valid in the DPS-DNG case withappropriate values for the material parameters.

It can be shown that at the interface of any given pair of semi-infiniteDPS-DNG media, either a TE surface wave or a TM surface wave, but notboth, may exist under the conditions:

$\begin{matrix}{{TE}\text{:}\left\{ {\begin{matrix}\left| \mu_{1} \middle| {< \left| \mu_{2} \right|} \right. \\\left| k_{1} \middle| {> \left| k_{2} \right|} \right.\end{matrix}\mspace{14mu}{or}\mspace{14mu}\left\{ {\begin{matrix}\left| \mu_{1} \middle| {> \left| \mu_{2} \right|} \right. \\\left| k_{1} \middle| {< \left| k_{2} \right|} \right.\end{matrix}\;,\;{{TM}\text{:}\left\{ {\begin{matrix}\left| ɛ_{1} \middle| {< \left| ɛ_{2} \right|} \right. \\\left| k_{1} \middle| {> \left| k_{2} \right|} \right.\end{matrix}\mspace{14mu}{or}\mspace{14mu}\left\{ \begin{matrix}\left| ɛ_{1} \middle| {> \left| ɛ_{2} \right|} \right. \\\left| k_{1} \middle| {< \left| k_{2} \right|} \right.\end{matrix} \right.} \right.}} \right.} \right.} & (26)\end{matrix}$

-   which are more stringent than the corresponding conditions (13)    obtained for SNG media, since in this case one must impose the    additional constraint β_(sw)>max (|k₁|,|k₂|). FIG. 6 presents the TE    and TM cases for β>k₂ in the same waveguides of FIGS. 4 b and 5 b    filled with a pair of DPS and DNG slabs. Here again the plots are    single-branched, since d₂ is single-valued, similar to the ENG-MNG    waveguide. FIG. 6 a illustrates normalized d₂ in terms of normalized    d₁ and β, in the region β>k₂, for TE modes in the waveguide filled    with a pair of DPS and DNG slabs with material parameters given as    in FIG. 4 b where k₁<k₂. As illustrated, for this choice of material    parameters, the DPS-DNG interface does not support a TE surface wave    or a TM surface wave. In FIG. 6 a,

$\frac{\left| \mu_{2} \middle| \sqrt{\beta^{2} - k_{1}^{2}} \right.}{\left| \mu_{1} \middle| \sqrt{\beta^{2} - k_{2}^{2}} \right.} > 1$for any value of β satisfying β>k₂>k₁, implying that for any given setof d₁ and β one can find a positive real-valued solution for d₂. It canbe observed that in the thin waveguide approximation, a TE mode exists,and the ratio d₁/d₂ again follows (16). Moreover, for any given d₁, thesolution for d₂ is monotonically decreasing with β in this region,suggesting that this DPS-DNG waveguide is mono-modal regardless of itstotal thickness in this range, similar to the case of ENG-MNG waveguidesdescribed above. In this case, however, generally the DPS-DNG waveguidemay also support a finite number of modes with β<k₂. In FIG. 6 a, d₁ mayattain any large value, still maintaining the mono-modal property inthis range, whereas d₂ is limited to a finite range to support thissingle mode.

FIG. 6 b illustrates the plot for the TM polarization in the sameDPS-DNG waveguide. In the thin waveguide limit, a no-cut-off TM mode issupported for all values of β in the range β>k₂, following (17) andcovering the other admissible values for γ complementary to the rangeshown in FIG. 5 b, and in agreement with FIG. 3. The plot in FIG. 6 b isonly available in the region where the inequality (10) is fulfilled,beyond which d₂ does not have a physical solution. At the boundary,where

$\begin{matrix}{{{\tanh\left( {\sqrt{\beta^{2} - k_{1}^{2}}d_{1}^{TM}} \right)} = \frac{\left| ɛ_{1} \middle| \sqrt{\beta^{2} - k_{2}^{2}} \right.}{\left| ɛ_{2} \middle| \sqrt{\beta^{2} - k_{1}^{2}} \right.}},} & (27)\end{matrix}$

-   the value of d₂ approaches +∞.

It is important to point out that in FIG. 6 b for a given set of d₁ andd₂, one may have two different modes supported in this range, due to thenon-monotonic behavior of the boundary (27) in the d₁−β plane. In otherwords, for a given set of material parameters and slab thicknesses, TMmodes in this case may possess two distinct solutions for β_(TM)>|k₂|.This anomalous behavior can be present only in waveguides filled with apair of materials that do not support any surface wave at theirinterface, as is the case in FIG. 6 b. However, as can be seen from FIG.6 a, it is clear that the TE case is different: for any d₁ the behaviorof d₂ in this case is monotonic with β, since the solution for d₂ startsfrom a non-zero value and decreases monotonically to zero when β→∞.

When the interface can support a surface wave, the situation isdifferent. FIG. 7 a illustrates normalized d₂ in terms of normalized d₁and β, in the region β>k₂, for TE modes in the waveguide filled with apair of DPS and DNG slabs with material parameters ε₁=3ε₀, μ₁=2μ₀,ε₂=−ε₀, μ₂=−3μ₀, where k₁<k₂. For this choice of material parameters,the DPS-DNG interface does support a TE surface wave. (The region β<k₂would be consistent with the results shown earlier). In this case,however, the asymptote generated by the surface wave modifies the plotsin the range β>k₂. The existence of a no-cut-off TE mode in the thinwaveguide approximation is clearly shown in FIG. 7 a near the regionwhere d₁→0 and d₂→0. Condition (10) again provides the allowable regionfor given sets of d₁ and β. The mono-modality in this range is alsoevident from FIG. 7 a.

The TM polarization for a DPS-DNG waveguide supporting a TM surface waveis shown in FIG. 7 b. Corresponding quantities for the TM modes in thewaveguide are filled with ε₁=−2ε₀, μ₁=−μ₀, ε₂=ε₀, μ₂=3μ₀, where k₁<k₂for which the DPS-DNG interface does support a TM surface wave. Herealso, the existence of the asymptotic behavior due to the surface waveprovides a certain specific variation for d₂ ^(TM). In particular, themono-modality in the range β>max(|k₁|,|k₂|) is present in this case.

D. Comparison of Modes in the ENG-MNG and DPS-DNG Waveguides

As noted above, several properties of the modes with β>max(|k₁|,|k₂|) inknown DPS-DNG waveguides resemble the modal characteristics in ENG-MNGwaveguides. The reason behind this similarity can be found in thetransmission-line analogy for ENG, MNG, DNG, and DPS media. In manyaspects DPS and DNG media behave similarly to SNG materials when anevanescent wave is considered inside these media.

One may suggest the following heuristic transformations, which map agiven SNG material (with parameters (ε, μ)) with propagation wave numberβ into an equivalent problem involving a DPS or DNG material withequivalent parameters (ε^(eq),μ^(eq)) and

$\begin{matrix}{{{\beta_{eq}\text{:}\left( {ɛ_{ENG},\mu_{ENG},\beta^{2}} \right)}->{\left( {ɛ_{DPS}^{eq},\mu_{DPS}^{eq},\beta_{eq}^{2}} \right) \equiv \left( {{- ɛ_{ENG}},\mu_{ENG},\left. {\beta^{2} + 2} \middle| k \right|^{2}} \right)}}{{{TE}\text{:}\left( {ɛ_{MNG},\mu_{MNG},\beta^{2}} \right)}->{\left( {ɛ_{DNG}^{eq},\mu_{DNG}^{eq},\beta_{eq}^{2}} \right) \equiv \left( {{- ɛ_{MNG}},\mu_{MNG},\left. {\beta^{2} + 2} \middle| k \right|^{2}} \right)}}\mspace{50mu}{\left( {ɛ_{ENG},\mu_{ENG},\beta^{2}} \right)->{\left( {ɛ_{DNG}^{eq},\mu_{DNG}^{eq},\beta_{eq}^{2}} \right) \equiv \left( {ɛ_{ENG},{- \mu_{ENG}},\left. {\beta^{2} + 2} \middle| k \right|^{2}} \right)}}{{{TM}\text{:}\left( {ɛ_{MNG},\mu_{MNG},\beta^{2}} \right)}->{\left( {ɛ_{DPS}^{eq},\mu_{DPS}^{eq},\beta_{eq}^{2}} \right) \equiv \left( {ɛ_{MNG},{- \mu_{MNG}},\left. {\beta^{2} + 2} \middle| k \right|^{2}} \right)}}} & (28)\end{matrix}$

In the above mapping from SNG materials to DPS and DNG materials, β²always maps to values greater than |k|². These transformations suggestthat for a given ENG-MNG waveguide with material parameters(ε_(ENG),μ_(ENG)) and (ε_(MNG),μ_(MNG)) and slab thicknesses d₁ and d₂,the behavior of the dispersion plot d₂ for any given set of d₁ and β(for TE and TM polarizations), is equivalent to the one of a suitablydesigned DPS-DNG waveguide, in the region β>max (|k₁|,|k₂|). It can beshown that in this case the mapping is given by the following conditionson the material parameters:

$\begin{matrix}{{\frac{\left| \mu_{DPS}^{eq} \right|}{\left| \mu_{DNG}^{eq} \right|} = \frac{\left| \mu_{ENG} \right|}{\left| \mu_{MNG} \right|}},{\frac{\left| ɛ_{DPS}^{eq} \right|}{\left| ɛ_{DNG}^{eq} \right|} = \frac{\left| ɛ_{ENG} \right|}{\left| ɛ_{MNG} \right|}},{\left| \left. ɛ_{DNG}^{eq}||\mu_{DNG}^{eq} \right. \middle| {- \left| ɛ_{DPS}^{eq}||\mu_{DPS}^{eq} \right|} \right. = {\quad{\left| \left. ɛ_{ENG}||\mu_{ENG} \right. \middle| {- \left| ɛ_{MNG}||\mu_{MNG} \right|} \right.,}}}} & (29)\end{matrix}$

-   when the slab thicknesses remain the same as d₁ and d₂. The    “equivalent” wave number for the guided mode in such a DPS-DNG    waveguide may be expressed in terms of the parameters of the    original ENG-MNG waveguide as:

$\begin{matrix}\begin{matrix}{\beta_{eq}^{2} = \left. {\beta^{2} + \omega^{2}} \middle| \left. ɛ_{ENG}||\mu_{ENG} \right. \middle| {+ \omega^{2}} \middle| \left. ɛ_{DPS}^{eq}||\mu_{DPS}^{eq} \right. \right|} \\{{= \left. {\beta^{2} + \omega^{2}} \middle| \left. ɛ_{MNG}||\mu_{MNG} \right. \middle| {+ \omega^{2}} \middle| \left. ɛ_{DNG}^{eq}||\mu_{DNG}^{eq} \right. \right|},}\end{matrix} & (30)\end{matrix}$

-   which is obviously greater than both k₁ ^(eq) and k₂ ^(eq) in the    DPS and DNG slabs. An analogous “inverse transformation” may map any    given DPS-DNG waveguide operating in the region β>max (|k₁|,|k₂|) to    an equivalent ENG-MNG waveguide.

From (29) and (30), some other analogies between the pairs of SNG layerswith the pairs of DPS and DNG layers may be summarized as follows:

-   1. ENG-ENG waveguides: for any β their dispersion relations are    equivalent to: a) (TE case) a DPS-DPS waveguide with β²>k_(i) ²    (i=1, 2) and thus no propagating mode is available; b) (TM case) a    DNG-DNG waveguide with β²>k_(i) ² (i=1, 2) and again no propagating    mode may exist.-   2. MNG-MNG waveguides: for similar reasons, no propagating mode may    exist in such waveguides.-   3. ENG-MNG waveguides: for any β their dispersion relations are    similar to the DPS-DNG waveguides with β²>k_(i) ² (i=1, 2) for both    polarizations. As shown above, these waveguides may be designed to    be mono-modal, regardless of their total thickness.-   4. DPS-ENG waveguides: when β²<k_(DPS) ² their dispersion diagrams    are equivalent to: a) (TE case) a DPS-DPS waveguide with k₁ ²<β²<k₂    ²; b) (TM case) a DPS-DNG waveguide with k₁ ²<β²<k₂ ². When    β²>k_(DPS) ², they become equivalent to: a) (TE case) an ENG-MNG    waveguide; b) (TM case) an MNG-MNG waveguide, both not supporting    any propagating mode.-   5. DPS-MNG, DNG-ENG, DNG-MNG waveguides: Corresponding results can    be obtained by using duality. In particular, the presence of a    no-cut-off TM mode in a standard DPS-DPS waveguide implies k₁    ²<β²<k₂ ², and this waveguide may then be regarded as a DPS-MNG    waveguide, which allows no cut-off solutions, following (17), for    any β<k_(DPS).

These and other analogies may be verified with the results set forthabove. A set of analyses employing this transmission-line analogy asapplied specifically to the waveguide geometries has been presented bythe present inventors in an article entitled“Distributed-circuit-element description of guided-wave structures andcavities involving double-negative or single-negative media,” Proc.SPIE: Complex Mediums IV: beyond Linear Isotropic Dielectrics, Vol.5218, San Diego, Calif., Aug. 4–5, 2003, pp. 145–155, the contents ofwhich are hereby incorporated by reference.

E. Guided Mode Waveguides Using at Least One SNG Material

Various properties of guided modes in parallel-plate waveguides filledwith pairs of layers made of any two of the lossless ENG, MNG, DPS, andDNG materials have been discussed above. It has been shown that,depending on the pairing the SNG materials and the choice of thematerial parameters, one may obtain modal features that differsignificantly from those of guided modes in conventional DPS-DPSwaveguides and DNG-DPS waveguides. Among several importantcharacteristics, the following are particularly significant: thepresence of TM and TE modes with no cut-off thickness in the limit ofthin DPS-DNG and ENG-MNG waveguides, the possibility of mono-modality inthick ENG-MNG waveguides and in DPS-DNG waveguides for slow modes withβ>max (|k₁|,|k₂|), and the presence of modes with wave number β greaterthan the wave numbers of both layers. These features suggestapplications of SNG materials (ENG and/or MNG) in the design ofultra-thin waveguides capable of supporting both TM and TE modes,single-mode thick fibers with less restriction and more flexibility onthe fiber thickness, very thin cavity resonators, and other noveldevices and components. The geometry is also not limited to parallelplate waveguides but also may include concentric waveguides where oneSNG material encircles the other SNG material, DPS or DNG material asillustrated in FIG. 1 b, and/or open waveguides without parallel plates.

In cases where the incident wave does not have an angle of incidencethat permits the creation of a surface wave or a guided mode in thebilayer material of the invention, the discussion of the followingsection sets forth the characteristic scattering properties of thebilayer material.

II. Resonance, Tunneling, and Transparency in a Pair of Slabs with atLeast One SNG Layer

A. Geometry

Consider a Cartesian coordinate system (x,y,z) with unit vectors{circumflex over (x)}, ŷ, and {circumflex over (z)}. Take atransverse-magnetic (TM) e^(jωt)-monochromatic plane wave in free spacewith its wave vector k₀={circumflex over (x)}k_(x)+{circumflex over(z)}√{square root over (k₀ ²−k_(x) ²)} in the x-z plane, withk₀=ω√{square root over (μ₀ε₀)}, and its magnetic and electric fieldvectors

H_(inc)^(TM)and

E_(inc)^(TM)given below:

$\begin{matrix}\begin{matrix}{H_{inc}^{TM} = {\hat{y}H_{0}{\mathbb{e}}^{{{- j}\; k_{x}x} - {j\sqrt{k_{0}^{2} - k_{x}^{2}}z}}}} \\{E_{inc}^{TM} = {\left( {{\hat{x}\frac{\sqrt{k_{0}^{2} - k_{x}^{2}}}{\omega\; ɛ_{0}}} - {\hat{z}\frac{k_{x}}{{\omega ɛ}_{0}}}} \right)H_{0}{\mathbb{e}}^{{{- j}\; k_{x}x} - {j\sqrt{k_{0}^{2} - k_{x}^{2}}z}}}}\end{matrix} & (31)\end{matrix}$

Assume the problem to be two-dimensional, i.e., all quantities areindependent of the y coordinate. Also assume the x-component of the wavevector, k_(x), to be a real quantity, and thus for the case of apropagating wave k_(x) ²≦ω²μ₀ε₀. If, however, k_(x) ²>ω²μ₀ε₀, the wavewill be evanescent, and the choice for the root square will be √{squareroot over (k₀ ²−k_(x) ²)}=−j√{square root over (k_(x) ²−k₀ ²)} to have adecaying wave along the +z direction.

Consider a pair of slabs of thicknesses d₁ and d₂, one made of alossless ENG material and the other of a lossless MNG material insertedin the region 0≦z≦d₁+d₂, as shown in FIG. 8. The regions 0 and 3 areconsidered to be free space. Take the TM incident wave to be the onegiven in Equation (31). Since in these ENG and MNG slabs με<0, k_(x)²>ω²με in each slab and thus waves in such layers are obviously alwaysevanescent. Therefore, the magnetic field vectors in the four regionscan be written as:

$\begin{matrix}\begin{matrix}{{H_{0}^{TM} = {\hat{y}H_{0}{{\mathbb{e}}^{{- j}\; k_{x}x}\left( {{\mathbb{e}}^{{- j}\sqrt{k_{0}^{2} - k_{x}^{2}}z} - {R^{TM}{\mathbb{e}}^{j\sqrt{k_{0}^{2} - k_{x}^{2}}z}}} \right)}}}\mspace{121mu}} & {{z < 0}\mspace{45mu}} \\{{H_{1}^{TM} = {\hat{y}H_{0}{{\mathbb{e}}^{{- j}\; k_{x}x}\left( {{C_{1 +}^{TM}{\mathbb{e}}^{{- \sqrt{k_{x}^{2} - k_{1}^{2}}}z}} + {C_{1 -}^{TM}{\mathbb{e}}^{\sqrt{k_{x}^{2} - k_{1}^{2}}z}}} \right)}}}\mspace{79mu}} & {0 < z < d_{1}} \\{H_{2}^{TM} = {\hat{y}H_{0}{{\mathbb{e}}^{{- j}\; k_{x}x}\left( {{C_{2 +}^{TM}{\mathbb{e}}^{{- \sqrt{k_{x}^{2} - k_{2}^{2}}}{({z - d_{1}})}}} + {C_{2 -}^{TM}{\mathbb{e}}^{\sqrt{k_{x}^{2} - k_{2}^{2}}{({z - d_{1}})}}}} \right)}}} & {d_{1} < z < d_{2}} \\{H_{3}^{TM} = {\hat{y}T^{TM}H_{0}{\mathbb{e}}^{{- j}\; k_{x}x}{\mathbb{e}}^{{{- j}\;\sqrt{k_{0}^{2} - k_{2}^{2}}{({z - d_{1} - d_{2}})}}\mspace{275mu}}}} & {{d_{1} + d_{2}} < z}\end{matrix} & {(32)\mspace{70mu}}\end{matrix}$where R^(TM) and T^(TM) are the reflection and transmissioncoefficients, the coefficient C's are the amplitude coefficients ofwaves in each of the slabs, normalized to the amplitude of the incidentwave, the subscripts (+) and (−) indicate the forward-decaying andbackward-decaying evanescent waves in each slab, and k_(n)=ω√{squareroot over (μ_(n))}√{square root over (ε_(n))}=k_(nr)−jk_(ni) with n=1, 2represents the wave number in each slab. The expression for theelectromagnetic fields in all four regions can be trivially obtainedfrom Maxwell's equations. Since these slabs are made of SNG materials,k_(ni)>0 and k_(ni)>>|k_(nr)|. The reflection and transmissioncoefficients and the coefficients C's can be obtained by requiring thetangential components of the electric and of the magnetic fields to becontinuous at the boundaries. The detailed expressions for thesecoefficients for the general 2-slab problem may be obtained, but asthese expressions are known to those skilled in the art and are verylong, they will not be included here.

To gain some insights into the behavior of the fields in these regions,instead the inventors present the plots of real and imaginary parts ofthe magnetic fields (H_(y) ^(TM) component) in these four regions forselected parameter values. FIG. 9 shows the magnetic field H_(y)behavior inside and outside the slabs for a sample pair of ENG-MNGslabs. In particular, FIG. 9 illustrates the real and imaginary parts ofthe normalized total transverse magnetic field as a function of zcoordinate, when a normally incident TM wave illuminates a pair oflossless ENG-MNG slabs (FIG. 9 a) and lossless MNG-ENG slabs (FIG. 9 b).Here the magnetic field intensity of the TM incident wave is assumed tobe unity, i.e., H₀=1 in Equation (31). In FIG. 9 a, the parameters arechosen to be

${ɛ_{1} = {{- 3}ɛ_{0}}},{\mu_{1} = {2\mu_{0}}},{ɛ_{2} = ɛ_{0}},{\mu_{2} = {{- 5}\mu_{0}}},{d_{1} = \frac{2\pi}{5{k_{1}}}},{d_{2} = \frac{2\pi}{5{k_{2}}}},{{\theta_{i} = 0};}$and in FIG. 9 b the two slabs are simply reversed in position. The valueof reflection, transmission, and C's coefficients are then found to beR=0.17+0.94j, T=−0.04+0.3j, C₁₊=1−1.19j, C¹⁻=−0.16+0.25j,C₂₊=−0.3+0.48j, C²⁻=0.01+0.04j in FIG. 9 a and R=0.4−0.87j,T=−0.04+0.3j, C₁₊=0.49+0.75j, C¹⁻=0.1+0.12j, C₂₊=0.56+0.59j,C²⁻=−0.06+0.03j in FIG. 9 b.

In FIG. 9 a, the slope of the term H_(y) has opposite signs on bothsides of the boundary z=d₁ between the slabs, in addition to the changeof slope sign at the boundary z=0. This is due to the fact that thetangential component of the electric field

$E_{x} = {\frac{- 1}{j\;\omega\; ɛ}\frac{\partial H_{y}}{\partial z}}$must be continuous at the boundaries, implying that

${{{{\frac{- 1}{j\;\omega\; ɛ_{0}}\frac{\partial H_{y}}{\partial z}}}_{z = 0^{-}} = {\frac{- 1}{j\;\omega\; ɛ_{1}}\frac{\partial H_{y}}{\partial z}}}}_{z = 0^{+}}.$Since in FIG. 9 a, ε ₀>0 and ε₁<0, the term

$\frac{\partial H_{y}}{\partial z}$thus has opposite signs on both sides of z=0 (and also on both sides ofz=d₁). As a result of this change of slope at z=d₁, the dominantbehavior of the magnetic field (for the TM case) in the two slabs isdifferent, i.e., if in the first slab the total magnetic field isdecreasing just before it gets to z=d₁, it will be increasing in thesecond slab just past z=d₁, or vice versa. FIG. 9 b shows the plot ofH_(y) for the case where the first slab is a MNG medium and the secondslab is an ENG medium. A similar observation can be made here with thechange of slope sign at z=d₁ and z=d₁+d₂.

This difference between the field behavior in the ENG and MNG parts ofthis bilayer structure obviously affects the transmission and reflectioncoefficients and, as will be shown in the next section, this can lead toan interesting anomalous tunneling, transparency and resonancephenomenon.

B. Equivalent Transmission-Line (TL) Model and Distributed CircuitElements for the ENG-MNG Bilayer

It is well known that considering the equivalent TL model for a TM planewave propagation in a homogeneous isotropic medium, one can write

$\frac{\partial E_{x}}{\partial z} = {{- j}\;\omega{\overset{\sim}{\mu}}_{eq}H_{y}}$and

$\frac{\partial H_{y}}{\partial z} - {j\;\omega{\overset{\sim}{ɛ}}_{eq}E_{x}}$where {tilde over (μ)}_(eq) and {tilde over (ε)}_(eq) are shorthand for

${\overset{\sim}{\mu}}_{eq} \equiv {{\mu\left( {1 - \frac{k_{x}^{2}}{\omega^{2}{\mu ɛ}}} \right)}\mspace{14mu}{and}\mspace{14mu}{\overset{\sim}{ɛ}}_{eq}} \equiv ɛ$for the TM case. (Using duality, one can find the corresponding termsfor the TE case.) With these expressions one can express the equivalentinductance per unit length L_(eq) and equivalent capacitance per unitlength C_(eq) for this TL model as:

$\begin{matrix}{{{L_{eq} \equiv {A_{1}{\overset{\sim}{\mu}}_{eq}}} = {A_{1}{\mu\left( {1 - \frac{k_{x}^{2}}{\omega^{2}\mu\; ɛ}} \right)}}},{{C_{eq} \equiv {A_{2}{\overset{\sim}{ɛ}}_{eq}}} = {A_{2}ɛ}}} & (33)\end{matrix}$where A₁ and A₂ are two positive constant coefficients that depend onthe geometry of the equivalent transmission line. Using the concept of{tilde over (μ)}_(eq) and {tilde over (μ)}_(eq), the inventors proposeappropriate equivalent TL models for waves, either propagating orevanescent waves, inside DPS, DNG, ENG, and MNG slabs. Such TL modelsfor the lossless case are listed in Table 1 for easy reference. Table 1illustrates effective TL models for TM waves in lossless DPS, DNG, ENG,MNG slabs. Both propagating and evanescent waves are considered inTable 1. In each case, a sketch of infinitesimal section of TL modelwith distributed series and shunt reactive elements per unit length ofthe line is shown. In addition, it is also indicated whether L_(eq) andC_(eq) is positive or negative, and whether β and Z are real orimaginary quantities. The symbols L-C, C-C, L-L, and C-L transmissionlines are used to indicate what effective reactive elements are used asthe distributed series (first symbol) and distributed shunt effectiveelements (second symbol), respectively, in the TL model. In assigningequivalent TL models for each of these cases in Table 1, the inventorstake into account the following points:

-   (i) if {tilde over (μ)}_(eq) happens to be a negative real quantity,    L_(eq) will be negative real, which conceptually translates into    “negative inductive reactance” jωL_(eq)=−jω|L_(eq)| at a given    frequency. This negative reactance can be effectively regarded as    the reactance of an equivalent “positive capacitance”, i.e.,

${{- j}\;\omega{L_{eq}}} = {- \frac{j}{\omega\; C_{eff}}}$for that given frequency. So in the TL analogy, whenever L_(eq)<0, itcan be thought of as an effective positive capacitance C_(eff). In sucha case, the TL model consists of series capacitance C_(eff) (instead ofconventional series inductance);

-   (ii) likewise, if {tilde over (ε)}_(eq)<0, C_(eq)<0, which can be    viewed as an effective shunt positive inductance L_(eff), i.e.,

${+ \frac{j}{\omega{\; C_{eq}}}} = {{+ j}\;\omega\;{L_{eff}.}}$In this situation, a shunt inductor (instead of a shunt capacitance)will appear in the TL model;

-   (iii) assuming k_(x) to be a real quantity. Thus, for a propagating    wave in a lossless medium, k_(x) ²≦ω²με, and for an evanescent wave    k_(x) ²>ω²με. Therefore, as mentioned earlier, waves in lossless ENG    and MNG slabs are obviously always evanescent, hence there is only    one single entry for each ENG and MNG case in Table 1;-   (iv) the inventors use the symbols L-C, C-C, L-L, and C-L    transmission lines to distinguish which reactive elements are used    as the distributed series (first symbol) and shunt elements (second    symbol), respectively, in the TL model; and finally-   (v) when the loss is present, one can always add the equivalent    distributed series resistance and shunt conductance per unit length    in the TL model given in Table 1.

TABLE 1 DPS (μ > 0, ε > 0) DNG (μ < 0, ε < 0) ENG (μ > 0, ε < 0) MNG (μ< 0, ε > 0) TM L_(eq) > 0 β ε

L_(eq) < 0 β ε

Not applicable, since Not applicable, since Propagating C_(eq) > 0 Z ε

C_(eq) < 0 Z ε

for k_(x) ε

, for k_(x) ε

, wavek_(x) ² < ω² με

k_(x) ² > ω² με for alllossless ENG k_(x) ² > ω² με for alllossless MNGTM L_(eq) < 0 β ε

L_(eq) > 0 β ε

L_(eq) > 0 β ε

L_(eq) < 0 β ε

Evanescent C_(eq) > 0 Z ε

C_(eq) < 0 Z ε

C_(eq) < 0 Z ε

C_(eq) > 0 Z ε

wavek_(x) ² > ω² με

It is noted that the equivalent C-L transmission line as the“left-handed” transmission line has already been considered in the priorart as a realization of one- and two-dimensional wave propagation in DNGmedia.

The paired ENG-MNG structure can now be viewed as the cascade oftransmission line segments with appropriate TL elements shown in Table1, representing the TM wave propagation in the ENG and MNG slabssandwiched between the two similar semi-infinite lines. Using thestandard TL formulation, one can find the following expression for thetransverse input impedance at the front face (z=0) of any genericbilayer followed by a uniform half space:

$\begin{matrix}{{Z_{i\; n}\left( {z = 0} \right)} = \frac{Z_{1}\left\{ {{j\; Z_{1}{{\tan\left( {\beta_{1}d_{1}} \right)}\left\lbrack {Z_{2} + {j\; Z_{0}\mspace{11mu}{\tan\left( {\beta_{2}d_{2}} \right)}}} \right\rbrack}} + {Z_{2}\left\lbrack {Z_{0} + {j\; Z_{2}{\tan\left( {\beta_{2}d_{2}} \right)}}} \right\rbrack}} \right\}}{{Z_{1}Z_{2}} + {j\left\{ {{Z_{0}Z_{1}{\tan\left( {\beta_{2}d_{2}} \right)}} + {Z_{2}{{\tan\left( {\beta_{1}d_{1}} \right)}\left\lbrack {Z_{0} + {j\; Z_{2}{\tan\left( {\beta_{2}d_{2}} \right)}}} \right\rbrack}}} \right\}}}} & (34)\end{matrix}$where the symbols Z's and β's indicate the characteristic impedance andwave number in each segment of this cascaded line and can be obtainedfrom Table 1 for different slabs.

The inventors are interested in exploring the possible conditions uponwhich one would get zero reflection from this structure, thus havingcomplete transmission of the TM wave through a pair of ENG-MNG slabs.Assuming for the moment that the first slab is ENG and the second isMNG, from Table 1 the lossless ENG slab the characteristic impedance ofthe equivalent L-L transmission line is purely inductive,

${Z_{1} \equiv Z_{ENG} \equiv {j\; X_{ENG}}} = \frac{\sqrt{k_{x}^{2} - {\omega^{2}\mu_{1}ɛ_{1}}}}{j\;\omega\; ɛ_{1}}$with X_(ENG)>0 since ε₁<0 and μ₁>0, whereas for the lossless MNG slabthe C-C line is purely capacitive, i.e.,

${Z_{2} \equiv Z_{MNG} \equiv {j\; X_{MNG}}} = \frac{\sqrt{k_{x}^{2} - {\omega^{2}\mu_{2}ɛ_{2}}}}{j\;\omega\; ɛ_{2}}$with X_(MNG)<0 since ε₂>0 and μ₂<0. In both slabs, the wave numbers areimaginary, i.e., β₁≡β_(ENG)=−j√{square root over (k_(x) ²−ω²μ₁ε₁)} andβ₂≡β_(MNG)=−j√{square root over (k_(x) ²−ω²μ₂ε₂)} since the wave insideeach slab is evanescent. The characteristics impedance Z₀ and the wavenumber β₀, of the semi-infinite segments of the TL, which representpropagating waves in the outside DPS region, are both real quantitiesexpressed as

$Z_{0} = \frac{\sqrt{{\omega^{2}\mu_{0}ɛ_{0}} - k_{x}^{2}}}{\omega\; ɛ_{0}}$and β₀=√{square root over (ω²μ₀ε₀−k_(x) ²)}. The zero-reflectioncondition Z_(in)(z=0)=Z₀ can be achieved if and only if:Z ₀(Z ₂ ² −Z ₁ ²)tan(β₁ d ₁)tan(β₂ d ₂)+jZ ₂(Z ₁ ² −Z ₀ ²)tan(β₁ d ₁)+jZ₁(Z ₂ ² −Z ₀ ²)tan(β₂ d ₂)=0  (35)Substituting the values of Z's and β's for the ENG and MNG slabs givenabove into Equation. (35) leads to the following expression:

$\begin{matrix}{{{{Z_{0}\left( {X_{ENG}^{2} - X_{MNG}^{2}} \right)}{\tanh\left( {{\beta_{ENG}}d_{1}} \right)}{\tanh\left( {{\beta_{MNG}}d_{2}} \right)}} + {j\left\lbrack {{{X_{MNG}\left( {Z_{0}^{2} + X_{ENG}^{2}} \right)}{\tanh\left( {{\beta_{ENG}}d_{1}} \right)}} + {{X_{ENG}\left( {Z_{0}^{2} + X_{MNG}^{2}} \right)}{\tanh\left( {{\beta_{MNG}}d_{2}} \right)}}} \right\rbrack}} = 0} & (36)\end{matrix}$The above condition will be satisfied for a pair of finite-thickness ENGand MNG slabs if and only if:

$\begin{matrix}\begin{matrix}{X_{ENG}^{2} = {X_{MNG}^{2}\mspace{14mu}{and}\mspace{14mu} X_{MNG}\mspace{14mu}{\tanh\left( {{\beta_{ENG}}d_{1}} \right)}}} \\{{= {{- X_{ENG}}{\tanh\left( {{\beta_{MNG}}d_{2}} \right)}}},}\end{matrix} & (37)\end{matrix}$leading to the conditions:X _(ENG) =−X _(MNG) and β_(ENG) d ₁=β_(MNG) d ₂.  (38)

A pair of lossless ENG and MNG slabs satisfying (38) gives rise to azero-reflection scenario, when it is sandwiched between two similar halfspaces. It is interesting to note that the above conditions in Equation(37) are the necessary and sufficient conditions for zero reflectionfrom any pair of “single-negative (SNG)” slabs sandwiched between twosimilar semi-infinite regions. Thus, for a given k_(x) if the parametersμ₁, ε₁ of the ENG slab, μ₂, ε₂ of the MNG slab and the thicknesses d₁and d₂ are chosen such that Equation (38) is fulfilled, a completetransmission of a wave can be had through this lossless ENG-MNG bilayerstructure, resulting in an interesting tunneling phenomenon. Theinventors name such an ENG-MNG pair a “matched pair” for the given valueof k_(x).

FIG. 10 illustrates a sketch of real and imaginary parts of thenormalized total transverse magnetic field (H_(y)) as a function of thez coordinate when a TM wave with 45° angle of incidence impinges on a“matched pair” of lossless ENG-MNG slabs (FIG. 10 a) and a “matchedpair” of lossless MNG-ENG slabs (FIG. 10 b). The parameters of theseslabs, which are chosen such that the zero-reflection conditions aresatisfied for the 45°-incident TM wave, are

${ɛ_{ENG} = {{- 3}\; ɛ_{0}}},{\mu_{ENG} = {2\mu_{0}}},{d_{ENG} = {{\frac{2\pi}{5{k_{ENG}}}\mspace{14mu}{and}\mspace{14mu} ɛ_{MNG}} = {2ɛ_{0}}}},{\mu_{MNG} = {{- 1.19}\mu_{o}}},{d_{MNG} = {\frac{2\pi}{5.28{k_{MNG}}}.}}$The reflection, transmission and C's coefficients are found to be: R=0,T=1, C₁₊=0.5∓0.42j, C¹⁻=0.5±0.42j, C₂₊=1.85±1.54j, C²⁻=0.14∓0.11j, wherethe upper (lower) sign refers to Case a (FIG. 10 b). The value of H_(y)at the front face of the pair (i.e., at z=0) is the same (both its realand imaginary parts) as that at the back face of the pair (i.e., atz=d₁+d₂), manifesting the complete tunneling of the incident wavethrough these lossless pairs, without any phase delay. The field valueswithin the ENG-MNG pair, however, can attain high values at theinterface between the two slabs (i.e., at z=d₁). Such unusual behaviorof the field inside and outside the ENG-MNG pair (or MNG-ENG pair) canbe justified by using the equivalent TL model, as will be shown below.FIG. 10 reveals the fact that the field variation inside the ENG-MNGpair can be different from that inside the MNG-ENG pair, even though theconditions of Equation 38 are the same for both matched pairs. In theformer, the real and imaginary parts of H_(y) inside the pair possessthe same sign (FIG. 10 a), while in the latter they have opposite signs(FIG. 10 b).

FIG. 11 illustrates the distribution of the real part of the Poyntingvector inside and outside the matched pair of ENG-MNG (FIG. 11 a) andMNG-ENG (FIG. 11 b), for the TM plane wave at

θ_(R = 0)^(TM) = 45^(∘)described with respect to FIG. 10. Here one can see the complete flow ofpower through the matched pair of slabs, highlighting the “completetunneling” phenomenon, transparency, and zero reflection property, aninteresting observation given the fact that each of the ENG and MNGslabs by itself would not have allowed a sizeable fraction of incidentpower to go through. Pairing the lossless ENG and MNG slabs thusprovides transparency for the incident wave at a particular angle, andleads to an interesting flow of the real part of the Poynting vectorinside the paired slabs.C. Characteristics of the Tunneling Conditions

Some of the salient features and characteristics of the zero-reflectionand complete tunneling conditions given in Equations (35)–(38) will bedescribed in this section.

1. Dependence on Material Parameters

From the derivation described in the previous section, it is clear thatif one exchanges the order of the slabs, i.e., instead of ENG-MNG pair,to have the MNG-ENG pair, the above conditions will remain unchanged.However, as shown in FIGS. 10 and 11, the field structure and the flowof the real part of the Poynting vector inside the two slabs will bedifferent.

The conditions given in Equations (35)–(38) are obtained for the ENG-MNGpair (or an MNG-ENG pair). However, when an ENG slab is next to anotherENG slab, the zero-reflection conditions will obviously never beachieved. When the ENG slab is juxtaposed even with a DPS or a DNG slab,the zero-reflection condition may not be satisfied either, for the casewhere the wave inside the DPS or DNG slab is assumed to be a propagatingwave. This is due to the fact that in such a case, Z₁ and β₁ of the ENGwould be purely imaginary, whereas Z₂ and β₂ of the DPS or DNG would bepurely real, and thus one cannot achieve a real Z_(in) in Equation (34).However, if k_(x) is chosen such that the TM wave inside the DPS or DNGslab is an evanescent wave, according to the equivalent TL models shownin Table 1, the DPS or DNG slab can be treated as an equivalent MNG orENG slab (for the TM mode), respectively, and a zero-reflectioncondition may, under certain conditions, be achievable.

The phenomenon of complete transmission through an inhomogeneous layerwith a particular permittivity profile and also through multilayeredstructures made of several layers of alternating plasma (with negativeepsilon) and conventional dielectric (with positive epsilon) materialshave been analyzed in the past by C.-H. Chen et al. However, thetunneling effect in those structures is due to the phenomenon of “leakyresonance” in which any positive-permittivity slab is placed between atleast two negative-permittivity layers. In the problem presented here,the tunneling phenomenon is different in that it occurs for a pair ofslabs consisting of only one ENG and one MNG layer.

The conditions derived in Equation (38) do not depend on the materialparameters of the two identical external regions, but only on theparameters of the ENG and MNG slabs. In fact, as will be shown in thenext section, these zero-reflection conditions are not due to thematching between the slabs with the external regions, but instead onlydue to the interaction of the ENG and MNG slabs with each other,resulting in a “resonance” phenomenon. However, if the two outsidesemi-infinite regions are filled by two different media, thezero-reflection condition will change, and will in general depend on theexternal parameters as well. It is important to note that even in thisgeneral case Equation (38) still represents a “transparency” condition,depending only on the internal interaction of the two slabs withthemselves. So when the conditions in Equation (38) are satisfied, thepaired slabs will become “transparent” to the incoming wave, and if thetwo outside media are the same, zero reflection will be achieved.

Moreover, it is interesting to note that the zero-reflection conditioncan also be satisfied by a pair of lossless DPS and DNG layers, if Z₁=Z₂and β₁d₁=−β₂d₂ for a given angle of incidence and polarization. This ispossible, since it is known from the teachings of Veselago that in DNGmedia the direction of phase velocity is opposite to the direction ofthe Poynting vector, so the condition β₁d₁=−β₂d₂ is achievable. Thecondition Z₁=Z₂ is also attainable. Such a DPS-DNG bilayer structurewill also be transparent to an incident wave with a specific angle andpolarization. The correspondence between the ENG-MNG pair and theDPS-DNG pair will be discussed in more detail below. In relation tothis, it is worth noting that Zhang and Fu have shown that the presenceof a DNG layer (or layers) can lead to unusual evanescent photontunneling when such DNG layers are next to conventional layers. Theircase can be considered as a special solution to Equation (37) above,since as shown in Table 1 the TL models for the evanescent waves in DPSand DNG layers are similar to those of MNG and ENG layers (for TM case)or ENG and MNG layers (for TE case), respectively.

2. Dependence on Slab Thicknesses

The conditions shown in Equation (38) do not restrict the sum of thethicknesses of the two slabs, d₁+d₂. One could thus have thick or thinlayers of lossless ENG and MNG materials as long as the above conditionsare satisfied in order to achieve transparency. When dissipation ispresent, the sum of the thicknesses can play a role, as will bediscussed below.

3. “Brewster-Type” Angle

The zero-reflection conditions given in Equation (38) in general dependon the value of k_(x). If the parameters of the ENG and MNG slabs arefirst chosen, one may be able to find a real value of k_(x) satisfyingEquation (38). If such a real k_(x) exists and if it satisfies theinequality k_(x) ²≦ω²μ₀ε₀, then it will be related to a particular angleof incidence of the TM wave for which the wave is “tunneled” through thelossless ENG-MNG bilayer structure completely and without anyreflection. This “Brewster-type” angle can be expressed as:

$\begin{matrix}{\theta_{R = 0}^{TM} \equiv {\arcsin\sqrt{\frac{ɛ_{1}{ɛ_{2}\left( {{ɛ_{2}\mu_{1}} - {ɛ_{1}\mu_{2}}} \right)}}{\mu_{o}{ɛ_{o}\left( {ɛ_{2}^{2} - ɛ_{1}^{2}} \right)}}}}} & (39)\end{matrix}$It should be remembered that in the above relation, ε₁<0, μ₁>0, ε₂>0,and μ₂<0 Obviously, an arbitrarily chosen set of such parameters for thepair of ENG and MNG slabs may not always provide one with an angleθ_(R=0) ^(TM) in the real physical space. In order to have such anangle, the following necessary condition should be fulfilled

$\begin{matrix}{\frac{1}{ɛ_{1}ɛ_{2}} < \frac{{ɛ_{2}\mu_{1}} - {ɛ_{1}\mu_{2}}}{\mu_{o}{ɛ_{o}\left( {ɛ_{2}^{2} - ɛ_{1}^{2}} \right)}} \leq {0\mspace{31mu}{for}\mspace{14mu}{TM}\mspace{14mu}{{case}.}}} & (40)\end{matrix}$As an aside, it is worth noting that the above condition coincides withthe one required for having a Zenneck wave at the interface betweensemi-infinite DPS and DNG media (and also semi-infinite ENG and MNGmedia).4. “Conjugate Matched Pair” of ENG-MNG Slabs

Of the infinite set of parameters satisfying Equation (38), theparticular set ε₁=−ε₂ μ₁=−μ₂, d₁=d₂ deserves special attention. Such apair of lossless ENG and MNG slabs is called herein the “conjugatematched pair” or “strictly matched pair,” in contradistinction with theterm “matched pair” defined earlier that referred to an ENG-MNG pairthat satisfies the general condition (38) for a specific value of k_(x).For the lossless conjugate matched pair, the reflection and transmissioncoefficients and the coefficient C's in the field expressions inside theslabs are simplified and expressed as:

$\begin{matrix}\begin{matrix}{{R^{TM} = 0},} & {{T^{TM} = 1},} & {{C_{1 \pm}^{TM} = {\frac{1}{2} \pm {j\frac{ɛ\sqrt{{\omega^{2}\mu_{0}ɛ_{0}} - k_{x}^{2}}}{2\sqrt{k_{x}^{2} - {\omega^{2}{\mu ɛ}}}}}}},} & {C_{2 \pm}^{TM} = {{\mathbb{e}}^{{\pm \sqrt{{k_{x}^{2} - {\omega^{2}{\mu ɛ}}}\;}}d}C_{1 \pm}^{TM}}} \\{{R^{TE} = 0},} & {{T^{TE} = 1},} & {{C_{1 \pm}^{TE} = {\frac{1}{2} \pm {j\frac{\mu\sqrt{{\omega^{2}\mu_{0}ɛ_{0}} - k_{x}^{2}}}{2\sqrt{k_{x}^{2} - {\omega^{2}{\mu ɛ}}}}}}},} & {C_{2 \pm}^{TE} = {{\mathbb{e}}^{{\pm \sqrt{{k_{x}^{2} - {\omega^{2}{\mu ɛ}}}\;}}d}C_{1 \pm}^{TE}}}\end{matrix} & (41)\end{matrix}$where ε≡ε₁=−ε₂, μ≡μ₁=−μ₂, and d≡d₁=d₂. For this case, thezero-reflection and complete tunneling through the slabs occurs for anyvalue of k_(x) and any angle of incidence (and for any polarizations,although only TM case is discussed here), hence T^(TM)=1. Moreover, noeffective phase delay due to the length d₁+d₂ is added to the wavepropagation, i.e., the phase of the transmitted wave at z=d₁+d₂ is thesame as the phase of the incident wave at z=0. Furthermore, thecoefficients C₁₊ ^(TM) and C¹⁻ ^(TM) in the first slab have the samemagnitude, which means that, according to Equation (32), the decayingand growing exponential terms in H_(y) have the same magnitude at z=0⁺.As the observation point moves through the first slab and approaches theinterface between the first and second slabs, the magnitude of H_(y)would be dominated by the growing exponential term. In the second slab,however, the magnitude of H_(y) is dominated by the decayingexponential, as the last two relations in Equation (41) require.Therefore, the field inside the conjugate matched pair of ENG-MNG slabsis predominantly concentrated around the interface between the twoslabs. This behavior can be seen from FIG. 12 a.

FIG. 12 a illustrates a sketch of real and imaginary parts of thenormalized total transverse magnetic field as a function of zcoordinate, when a normally incident TM wave impinges on a “conjugatematched pair” of lossless ENG-MNG slabs, while FIG. 12 b illustrates thedistribution of the real and imaginary part of the normalized Poyntingvector inside and outside of this structure. The normalization is withrespect to the value of the Poynting vector of the incident wave. It isnoted that the real part of the normalized Poynting vector is uniformand equals unity through the paired slabs, implying the completetunneling of the incident wave, whereas the imaginary part of thePoynting vector is only present inside the pair and has its peak at theinterface between the two slabs. The parameters of the ENG and MNGslabs, which are chosen such that the conjugate matched pair conditionsare satisfied, are ε_(ENG)=−3ε₀, μ_(ENG)=2μ₀, ε_(MNG)=3ε₀, μ_(MNG)=−2μ₀and

$d_{1} = {d_{2} = {\frac{2\pi}{5{k_{1}}}.}}$The reflection, transmission and C's coefficients for the pair are foundto be R=0, T=1, C_(1±)=0.5∓0.61j, C₂₊=1.76+2.15j, C²⁻=0.14−0.17j.

The real and imaginary parts of the Poynting vector for a wave tunnelingthrough these conjugate matched pair of ENG and MNG slabs is shown inFIG. 12 b, where the case of a normally incident wave is considered.From FIG. 12 b it can be seen the real part of the Poynting vector isuniform and equals unity through the structure, indicating the completetunneling phenomenon. The imaginary part of the Poynting vector, on theother hand, is zero outside the paired slabs, it is only present insidethe slab, and has its peak at the interface between the two slabs. Thisexhibits the presence of stored energy in these paired slabs, which, aswill be explained below, can be regarded as a “resonance” phenomenon.

5. Variation in Angle of Incidence for the Matched Pair of ENG-MNG Slabs

For a given set of parameters for the ENG and MNG slabs, the generalmatched pair condition for zero reflection and transparency may besatisfied if the TM incident wave can have a specific incident angle,

θ_(R = 0)^(TM),given in Equation (39). This implies that when an ENG-MNG pair isdesigned to be transparent for a TM wave with a specific angle ofincidence, this pair will not be transparent to other angles ofincidence. Recall that the zero reflection and transparency conditionfor the more specific conjugate pair is independent of the angle ofincidence. The sensitivity of reflection coefficient for the generalmatched pair to this angular variation will now be explored. A variationδk_(x) in the transverse wave number k_(x) causes a perturbation in thezero-reflection conditions, which can be expressed, to a first-orderapproximation, as

${{j\; X_{ENG}} + {j\; X_{MNG}}} \cong {\frac{j\; k_{x}\delta\; k_{x}}{\omega^{2}X_{ENGm}}\left( {\frac{1}{ɛ_{1}^{2}} - \frac{1}{ɛ_{2}^{2}}} \right)}$and

${{{\beta_{ENG}d_{1}} - {\beta_{MNG}d_{2}}} \cong \frac{k_{x}\delta\;{k_{x}\left( {d_{2}^{2} - d_{1}^{2}} \right)}}{\beta_{ENGm}d_{1}}},$where X_(ENGm), β_(ENGm) and β_(MNGm) are the values satisfying thematched condition (38), i.e. for δk_(x)=0. The reflection sensitivity onthe angular variation, therefore, increases with k_(x).

Moreover, the reflectivity increases with the difference between theconstitutive parameters in the two media and the total thickness of thestructure. FIG. 13 shows the magnitude of the reflection coefficient forthe ENG-MNG pair with several sets of parameters designed to make thepair transparent at

$\theta_{i} = {\frac{\pi}{4}.}$As illustrated in FIG. 13, the reflection coefficient is sensitive tovariation of the angle of incidence. The magnitude of the reflectioncoefficient from matched pairs of lossless ENG-MNG is plotted as afunction of angle of incidence. An ENG slab with parametersε_(ENG)=−3ε₀, μ_(ENG)=2μ₀ and

$d_{ENG} = \frac{2\pi}{5{k_{ENG}}}$for FIG. 13 a and

$d_{ENG} = \frac{4\pi}{5{k_{ENG}}}$for FIG. 13 b is first selected. Then, the parameters of the MNG slabare chosen such that the pair satisfies the zero-reflection condition(Equation (38)) for the TM wave with 45° angle of incidence. Since thereare two relations in Equation (38), but there are three parameters(ε_(MWG), μ_(MNG), d_(MNG)) to determine for the MNG slab, there is onedegree of freedom. As a result, in principle, for a given ENG the choiceof MNG is not unique in order to form a matched ENG-MNG pair. A familyof curves for several pairs of matched ENG-MNG slabs is shown. For eachpair, the reflection coefficient is then evaluated as a function ofangle of incidence. The values of parameters for the MNG slab are shownnear each plot (the values of the permittivity and permeability of theMNG slab are shown with respect to ε₀ and μ₀). The variation of thereflection coefficient with angle of incidence is less sensitive forthinner slabs (FIG. 13 a) than for the thicker ones (FIG. 13 b).

When the media parameters are chosen closer to the conjugate matchedpair conditions, the reflectivity remains low for a wider set of angles,whereas for larger values of

${{\frac{1}{ɛ_{1}^{2}} - \frac{1}{ɛ_{2}^{2}}}},$the reflectivity increases as the angle of incidence deviates from thedesign angle

θ_(R = 0)^(TM).

Such dependence on the difference between the constitutive parameterssaturates for large ε₂, as seen from FIG. 13 and the first-orderapproximation. The reflectivity will depend more on the total thicknessof the structure. As can be seen from FIG. 13, the reflectivity from athinner structure (FIG. 13 a) is less sensitive to the angular variationthan that from the thicker pair (FIG. 13 b).

6. Presence of Material Loss

In finding the matched pair conditions given in Equations (36)–(38) andthe conjugate matched pair conditions, the inventors considered losslessENG and MNG slabs. Obviously, with the presence of loss, perfecttransparency and zero reflection is not achievable due to the mismatchbetween the paired slabs and the outside region, as well as theabsorption in the materials. It is important to explore the sensitivityof the wave tunneling phenomenon on the value of ε_(i) or μ_(i). Imaginea conjugate matched pair of ENG-MNG slabs, with complete tunneling,i.e., then the zero reflection for the lossless case can be achieved.FIG. 14 shows how the reflection and transmission coefficients vary withε_(i) and/or d. In particular, FIG. 14 illustrates the magnitude of thereflection coefficient (FIG. 14 a) and the transmission coefficient(FIG. 14 b) for the conjugate matched pair of ENG-MNG slabs shown inFIG. 12 a when the loss mechanism is introduced in the permittivity ofboth slabs as the imaginary part of the permittivity. (Normal incidenceis assumed here.) It is noted that the sensitivity of the reflection andtransmission coefficients to the presence of loss (i.e., on ε_(i))depends on the value of the thickness d; i.e., the reflection andtransmission coefficients will become more sensitive to variations withrespect to ε_(i) as the value of d becomes larger.

In FIG. 14, it is assumed that μ_(i)=0, while ε_(i) is allowed to benon-zero. (Analogous results are obtained if ε_(i)=0, and μ_(i)≠0) Notethat for ε_(i)=0, the reflection coefficient is zero for all values ofd, representing the case of a conjugate matched pair. However, whenε_(i) becomes non-zero, the reflection coefficient may attain non-zerovalues, and the sensitivity of the reflection coefficient on ε_(i)depends on the value of d: the larger the value of d, the more sensitivethe reflection coefficient will be with respect to ε_(i). This would beexpected from physical arguments. For small values of d, the reflectioncoefficient is not too sensitive to the presence of small ε_(i). Inaddition, one can also determine how the transmission of a wave throughthe pair is affected by the loss.

FIG. 15 illustrates for the TM case the behavior of the total magneticfield inside and outside of the ENG-MNG pair, when ε_(i) and μ_(i) areallowed to be non-zero. In particular, FIG. 15 illustrates the effect ofloss on the field distribution in a conjugate matched ENG-MNG pairexcited by a normally incident plane wave. FIG. 15 shows thedistribution of the real and imaginary parts of the total transversemagnetic field inside and outside of the conjugate matched pair ofENG-MNG slabs considered in FIG. 12 a, when the loss is present in theform of the imaginary part of the permittivity in both slabs. Obviously,for larger values of ε_(i) and μ_(i), more absorption occurs in thestructure, resulting in lower values of the transmitted wave.

D. Resonance in the ENG-MNG Bilayer

As listed in Table 1, the evanescent TM wave propagation in the ENG andMNG slabs can be modeled as equivalent L-L and C-C transmission lines,respectively. Therefore, the ENG-MNG pair can be modeled using the TL,as shown in FIG. 16 a. In FIG. 16 a, this pair is shown as the cascadedL-L and C-C transmission lines in the range 0<z<d₁ and d₁<z<d₁+d₂,respectively. The two semi-infinite regions of free space, z<0 andz>d₁+d₂, can of course be modeled as the standard L-C transmission linesfor the case of a propagating TM wave.

FIG. 16 illustrates the equivalent transmission line models withcorresponding distributed series and shunt elements, representing the TMwave interaction with a pair of ENG-MNG slabs (FIG. 16 a), and a pair ofDPS-DNG slabs (FIG. 16 b). In each section, the TL consists of manyinfinitesimally thin cells containing series and shunt elements. Asshown in Table 1, the choice of such equivalent series and shuntelements in the TL model depends on the material parameters in eachslab, the polarization of the wave (here only TM polarization isconsidered), and whether the wave is propagating or evanescent.

It is interesting to observe how the TL section between z=0 and z=d+d₂affects the propagation through the line from left to right. Firstassume that the L-L line and C-C line are divided into many smallinfinitesimal segments. The number of such infinitesimal segments istaken to be N=N′>>1, thus the length of each segment in the L-L and C-Clines is d₁/N and d₂/N′, respectively. Each segment has certain amountsof series impedance and shunt admittance, which can be obtained bymultiplying, respectively, the segment length by the series impedanceper unit length and shunt admittance per unit length. Looking at thenodes N−1 and N′−1, and referring to FIG. 16 a, the total impedancebetween these two nodes can be written as:

$\begin{matrix}{Z_{{N - 1},{N^{\prime} - 1}} = {{{j\omega}\; L_{{eq}\; 1}\frac{d_{1}}{N}} + {\frac{1}{{j\omega}\; C_{{eff}\; 2}}{\frac{d_{2}}{N}.}}}} & (42)\end{matrix}$

As described earlier, C_(eff2) is defined as

$C_{{eff}\; 2} \equiv {- {\frac{1}{\omega^{2}L_{{eq}\; 2}}.}}$By substituting the values of L_(eq1) and L_(eq2) for the TM case theabove equation can be re-written as:

$\begin{matrix}{Z_{{N - 1},{N^{\prime} - 1}} = {{\frac{j\; A_{1}\omega}{N}\left\lbrack {{\left( {1 - \frac{k_{x}^{2}}{\omega^{2}\mu_{1}ɛ_{1}}} \right)\mu_{1}d_{1}} + {\left( {1 - \frac{k_{x}^{2}}{\omega^{2}\mu_{2}ɛ_{2}}} \right)\mu_{2}d_{2}}} \right\rbrack}.}} & (43)\end{matrix}$Taking into account the conditions given in Equation (38) for a matchedpair and considering the fact that μ₁ε₁<0 and μ₂ε₂<0 after somemathematical manipulations, Z_(N−1,N′−1)=0 is obtained. This impliesthat the series reactive elements between the nodes N−1 and N′−1 are inresonance, and thus these two nodes have the same voltage, i.e.,V_(N−1,N′−1)≡V_(N−1)−V_(N′−1)=0. From this, one can assert that theshunt element L_(eff1) at node N−1 (i.e. between node N−1 and theground) and the shunt element C_(eq2) at node N′−1 are now in parallel.The total admittance of these two parallel shunt reactive elements is:

$\begin{matrix}{Y_{{N - {1/N^{\prime}} - 1},\mspace{11mu}{ground}} = {{\frac{1}{{j\omega}\; L_{{eff}\; 1}}\frac{d_{1}}{N}} + {{j\omega}\; C_{{eq}\; 2}{\frac{d_{2}}{N}.}}}} & (44)\end{matrix}$

Following similar steps and considering that

${L_{{eff}\; 1} \equiv {- \frac{1}{\omega^{2}\; C_{{eq}\; 1}}}},{Y_{{N - {1/N^{\prime}} - 1},\mspace{11mu}{ground}} = 0},$which implies that the two shunt elements are in resonance.Consequently, the current flowing in the inductive element L_(eq1)between the nodes N−2 and N−1 is the same as the current flowing in thecapacitive element C_(eff2) between the nodes N′−1 and N′−2. These twoelements are then effectively in series, since Y_(n−1/n′−1,ground)=0.Repeating the above steps it can be shown that these two elements are inresonance as well, i.e., Z_(N−2,N′−2)=0, resulting inV_(N−2,N′−2)≡V_(N−2)−V_(N′−2)=0. Following this procedure away from themiddle node N≡N′ towards the end nodes 0 and 0′, such resonance behavioroccurs for every pair of series and shunt elements, and thus the currentand voltage at the node 0 are the same at those at node 0′, i.e.:V _(z=0) =V _(z=d) ₁ _(+d) ₂ and I _(z=0) =I _(z=d) ₁ _(+d) ₂   (45)

From this, one can conclude that the input impedance at z=0 (lookinginto the right in FIG. 16 a) is the same as the input impedance atz=d₁+d₂ (looking into the right). Therefore, the segment of the TLbetween z=0 and z=d₁+d₂, which represents the matched pair of losslessENG and MNG slabs, is in resonance and has become “transparent” to theincoming wave that is effectively “tunneling” through this segment withno effective phase change. Although the lossless ENG or MNG slab byitself does not allow the perfect complete tunneling of the incomingwave through it, when an ENG slab is juxtaposed with an MNG slab withthe properly selected set of parameters, a resonant structure whichprovides transparency and zero reflection to the incoming wave isobtained. If the slab parameters are chosen to have a conjugate matchedlossless pair, then an incident wave with any angle of incident andpolarization can tunnel through the pair. From a TL and circuit elementpoint of view, one can also see that with an individual one of the L-Lor C-C transmission lines, the current and voltage will decay along sucha line. However, when the L-L and the C-C segments are joined torepresent the ENG-MNG pair, a resonant structure in which the currentand voltage behave quite differently is found.

In order to intuitively understand and interpret the field behaviorinside the ENG-MNG pair, the current in each series element in theresonant segment between z=0 and z=d₁+d₂ in FIG. 16 a is evaluated.Considering the fact that at z=0 there is no reflection, and thusV_(z=0)/I_(z=0)=Z₀, where Z₀ is the characteristic impedance of the TLbefore the point z=0, the following expression for the current in theseries elements between nodes M and M+1, where 0≦M≦N−1 is found:I _(M/M+1) =I _(M/M+1,r) +jI _(M/M+1,i)where

$\begin{matrix}{{I_{{{M/M} + 1},r} = {\frac{I_{z = 0}}{L_{{eff}\; 1}^{M}2^{M + 1}}\left\lbrack {{\left( {1 + \frac{\sqrt{L_{{eq}\; 1}}}{\sqrt{L_{{eq}\; 1} + {4L_{{eff}\; 1}}}}} \right)\alpha_{1}^{M}} + {\left( {1 - \frac{\sqrt{L_{{eq}\; 1}}}{\sqrt{L_{{eq}\; 1} + {4L_{{eff}\; 1}}}}} \right)\alpha_{2}^{M}}} \right\rbrack}}{I_{{{M/M} + 1},i} = {\frac{I_{z = 0}}{2^{M}L_{{eff}\; 1}^{M}}\frac{Z_{o}}{\omega\; L_{{eq}\; 1}}\frac{\sqrt{L_{{eq}\; 1}}}{\sqrt{L_{{eq}\; 1} + {4L_{{eff}\; 1}}}}\left( {a_{1}^{M} - a_{2}^{M}} \right)}}} & (46)\end{matrix}$with a₁ and a₂ being shorthand for

$a_{\underset{2}{1}} \equiv {L_{{eq}\; 1} + {{2L_{eff1}} \mp {\sqrt{L_{{eq}\; 1}\left( {L_{{eq}\; 1} + {4L_{{eff}\; 1}}} \right)}.}}}$If for the special case where L_(eq1)=L_(eff1)≡L, then the aboveequation can be simplified as:

$\begin{matrix}{{I_{{{M/M} + 1},r} = {I_{z = 0}\;\frac{{\left( {3 + \sqrt{5}} \right)^{M}\left( {\sqrt{5} + 1} \right)} + {\left( {3 - \sqrt{5}} \right)^{M}\left( {\sqrt{5} - 1} \right)}}{2^{M + 1}\sqrt{5}}}}{I_{{{M/M} + 1},i} = {I_{z = 0}\;\frac{\left( {3 + \sqrt{5}} \right)^{M} - \left( {3 - \sqrt{5}} \right)^{M}}{2^{M}\sqrt{5}}\frac{Z_{o}}{\omega\; L_{{eq}\; 1}}}}} & (47)\end{matrix}$

The corresponding expressions for the current in series elements betweenthe consecutive nodes M′ and M′+1, where 0′<M′<N′−1 is a node in the C−CTL section, can also be found by simply making the followingsubstitutions L_(eq1)→1/ω²C_(eff2), L_(eff1)→1/ω²C_(eq2) in Equations(46) and (47). The currents I_(M/M+1) and I_(M′/M′+1) increase withpower of M, when M increases from 0 to N−1. The same considerations canbe made for the corresponding expressions for the voltages at the nodesM and M′. Their behavior is very similar. Consequently, in thisequivalent circuit model, it may be observed that the current andvoltage distributions within this resonant segment are concentratedaround the node N≡N′ and then they decrease in magnitude as oneapproaches to the edges of the segment, i.e., towards the nodes 0 and0′. This is obviously consistent with what was obtained for the fieldbehavior using the wave theory, as shown in FIGS. 10 and 12.

It is worth noting that the resonant behavior of the ENG-MNG segment ofthe line can also be interpreted in terms of the impedance mismatchbetween the two TL sections at the node N≡N′. As listed in Table 1, thecharacteristic impedance of the L-L line representing the ENG slab (forthe TM mode) is a positive imaginary quantity, Z_(L-L)=jX_(L-L) withX_(L-L)>0 while the one for the C-C line for the MNG slab is a negativeimaginary quantity, Z_(C-C)=jX_(C-C) with X_(C-C)<0. If one treats theevanescent wave in the ENG slab as an “incoming” wave impinging on theboundary between the ENG and MNG slabs, the Fresnel “reflectioncoefficient” for such an incident evanescent wave can then be written asR=(Z_(C-C)−Z_(L-L))/(Z_(C-C)+Z_(L-L)). When the ENG-MNG pair is amatched pair, according to the conditions (38), Z_(L-L)=−Z_(C-C),resulting in an infinitely large reflection coefficient! This should notcause any concern, because (1) this is a reflection coefficient betweenan “incident” evanescent wave and “reflected” evanescent wave, whicheach by itself does not carry any real power; and (2) the “singular”nature of this reflection coefficient implies that there is a resonant“natural” mode for this segment of the line acting effectively as a“cavity resonator”.

E. Correspondence Between the ENG-MNG Pair and the DPS-DNG Pair

Fredkin and Ron have shown that a layered structure with alternatingslabs of negative-epsilon and negative-mu materials may effectivelybehave as a DNG material, because the effective group velocity in such astructure would be antiparallel with the effective phase velocity. Here,using the TL model, the inventors present a different analogy betweenthe ENG-MNG pair and the DPS-DNG pair. The transmission line model shownin FIG. 16 a can be modified to represent the TM wave interaction with alossless DPS-DNG bilayer structure. Using the information given in Table1, one can model the DPS-DNG pair as the transmission line shown in FIG.16 b. One can immediately see that, like the case of ENG-MNG pair, ifthe parameters of the DPS-DNG pair are chosen such that thezero-reflection conditions are satisfied, the resonance phenomenonbetween the series reactive elements and between the shunt reactiveelements in infinitesimal sections of the TL will occur. As a result,one can again have V_(z=0)=V_(z=d) ₁ _(+d) ₂ and I_(z=0)=I_(z=d) ₁ _(+d)₂ , suggesting that the matched pair of lossless DPS and DNG slabs canbe in resonance and may become “transparent” to the incoming wave. Inthis sense, the ENG-MNG pair may act in a similar manner as the DPS-DNGpair. However, one should remember that the behavior of the wavepropagation within these two pairs is different: in the ENG-MNG pair,the fields are the sum of evanescent waves, whereas in the DPS-DNG pair,one can have propagating waves. Since in the small-argumentapproximation, trigonometric sinusoidal and hyperbolic sinusoidalfunctions may appear somewhat similar, for a short range of distance theevanescent field function may approximately resemble those of thepropagating wave. One can then anticipate that a “thin” ENG-MNG pair mayessentially function like a thin DPS-DNG pair. This point is pictoriallyillustrated in FIG. 17.

With respect to FIG. 17, one may imagine cascaded “thin” layers ofidentical lossless ENG-MNG pairs. The layers are assumed to beelectrically thin with thicknesses Δd₁ and Δd₂. The equivalent cascadedpair, L-L and C-C, transmission line model of this structure is alsoshown in FIG. 17. It can be seen from this model that if the layers areassumed to be thin enough, these cascaded pairs can also be viewed ascascaded pairs of L-C and C-L lines, thus representing cascaded pairs ofDPS-DNG layers. Specifically, notice that each segment ofL_(eq1)−L_(eff1) is sandwiched between the two segments ofC_(eff2)−C_(eq2), and vice versa. Therefore, for thin layers thegrouping of L_(eq1)−C_(eq2) and C_(eff2)−L_(eff1) together can beassumed instead, representing thin DPS and DNG layers, respectively. Ifthe matched pair conditions, Equation (38), for the ENG-MNG pair aresatisfied, one would get the tunneling effect for the entire set ofcascaded ENG-MNG pairs, since all these pairs behave as resonantstructures individually. If the matched pair conditions are fulfilled,the same can be said about the set of cascaded pairs of DPS-DNG slabsfor the evanescent tunneling.

As noted above, analogous behaviors can also be exploited in guided-wavestructures. As was the case with the paired parallel DPS-DNG layersinserted in a parallel-plate cavity or waveguide structure, the matchedENG-MNG paired parallel slabs could also support a resonant mode whenthey are placed between two parallel metallic walls. Furthermore, thephase of the field at the back face of this matched pair is also thesame as the phase at the front face.

F. “Ideal” Image Displacement and Image Reconstruction

In this section, a pair of lossless ENG-MNG slabs that are conjugatematched for a given fixed frequency will be considered. In front of thispair, the inventors put an object, e.g., a line source as shown in FIG.18. The field distribution at the object plane can in general beexpanded in terms of spatial Fourier components with spatial wavenumberparameters k_(x) and k_(y). Here, for the sake of simplicity, it isassumed that the object is independent of the y coordinate and has a 1-DFourier expansion in terms of spatial components with wavenumber k_(x)only, where −∞<k_(x)<+∞. Since the lossless ENG-MNG pair is conjugatematched, each of the spatial Fourier components, propagating as well asevanescent waves in the outside region, will in principle tunnel throughthe pair, and they show up at the exit face with the same correspondingvalues as their values at the entrance face. This means that an observeron the back side of the paired slab will see the object as though itwere displaced and seated closer to the observer by the amount d₁+d₂. Infact, conceptually all of its spatial Fourier components are preserved.In the absence of the ENG-MNG pair, if the distance between the objectand the observer is assumed to be D, the observer for large enough Dwill only receive the propagating waves from the object, and theevanescent wave portion of the Fourier decomposition will be negligibleat the observation plane. However, if the conjugate matched losslessENG-MNG pair is inserted in the region between the object and theobserver, the apparent location of the object will be at the distanceD−2d, which may provide near-field observation of the objects withtheoretically all spatial Fourier components (propagating and evanescentcomponents in the outside region) present, i.e., with its originalresolution intact. This can provide an interesting application of suchpaired slabs in image reconstruction and resolution enhancement.

Also, an analogous matched pair of lossless DPS-DNG slabs would also“preserve”, and allow “tunneling”, of the evanescent waves as theyinteract with such a pair, similar to what Pendry has analyzed for a DNGslab surrounded by a DPS medium. However, unlike the case of an ENG-MNGpair, propagating waves can exist in the DPS-DNG pair, and owing to theanomalous negative refraction at the interfaces between DNG and DPSmaterials, a focusing effect occurs leading to a real image. Such afocusing effect for a DNG slab has already been suggested and studied byPendry in the particular case in which the DPS slab material is taken tobe the same as the outside region, i.e., free space. In general, for aDPS-DNG pair a real image of an object can be formed, whereas for anENG-MNG pair, a virtual image can be obtained. This behavior isillustrated in FIG. 18. As illustrated in FIG. 18, when such a pair ofENG-MNG slabs is inserted between the object (on the left) and theobserver (on the right), that is, at a distance D away from the object,the virtual image of the object appears closer to the observer, at thedistance D−2d, with ideally all its spatial Fourier components present,i.e., with its original resolution intact. Those skilled in the art willappreciate that this technique permits displacement of an image wherebynear field imaging may be accomplished even though an imaging probe isplaced further away from the object of interest, assuming that theENG-MNG layers are transparent.

Although implementations of the invention have been described in detailabove, those skilled in the art will readily appreciate that manyadditional modifications are possible without materially departing fromthe novel teachings and advantages of the invention. For example, thoseskilled in the art will appreciate other possible applications of thisinvention including possible use in design of stealthy objects,miniaturization of cavities and waveguides and related devices andcomponents, in design of thin Fabry-Perot technology, and reduction ofreflection in solar energy transducers, to name a few. Any suchmodifications are intended to be included within the scope of theinvention as defined in the following claims.

1. A structure for use in waveguiding or scattering of waves, thestructure comprising first and second adjacent layers, the first layercomprising an epsilon-negative (ENG) material or a mu-negative (MNG)material, and the second layer comprising either (1) a double-positive(DPS) material, (2) a double-negative (DNG) material, (3) an ENGmaterial when said first layer is an MNG material, or (4) a MNG materialwhen said first layer is an ENG material.
 2. The structure of claim 1,wherein the first and second layers are parallel so as to form awaveguide.
 3. The structure of claim 2, wherein said first and secondlayers are bounded by metal plates in a direction parallel to apropagation direction of a guided wave whereby said waveguide is aclosed waveguide.
 4. The structure of claim 3, wherein the first layerhas a thickness d₁, permittivity ε₁, and permeability μ₁, and the secondlayer has a thickness d₂, permittivity ε₂, and permeability μ₂, wherebyif the thicknesses d₁ and d₂ are assumed to be very small in a directionperpendicular to the propagation direction of the guided wave and theparameters μ₁, μ₂, ε₁, ε₂ satisfy the following equations:$\begin{matrix}{{{\gamma\bullet} - {\frac{\mu_{2}}{\mu_{1}}\mspace{14mu}{for}\mspace{14mu}{TE}\mspace{14mu}{mode}}},{and}} \\{{{\beta_{TM}\bullet} \pm {\omega\sqrt{\frac{{\mu_{1}\gamma} + \mu_{2}}{{\gamma/ɛ_{1}} + {1/ɛ_{2}}}}\mspace{11mu}{for}\mspace{14mu}{TM}\mspace{14mu}{mode}}},}\end{matrix}$ where γ is defined as d₁/d₂ and β_(TM) is a wave number ina TM mode that depends on the ratio of layer thicknesses, d₁/d₂, not onthe total thickness (d₁+d₂), then the TM and TE modes have no cut-offthickness below which the TE mode may not propagate.
 5. The structure ofclaim 4, wherein at β=0 d₁ and d₂ are selected such that d₁/d₂≈−μ₂/μ₁,thereby forming a cavity resonator in said closed waveguide that isindependent of d₁+d₂.
 6. The structure of claim 2, wherein the firstlayer comprises said ENG material, said ENG material having a thicknessd₁, and the second layer comprises said MNG material, MNG materialhaving a thickness d₂, and wherein said structure provides a single modewhen d₁ and d₂ are selected to be multiples of a wavelength of a guidedwave whereby the single mode is effectively independent of a value ofd₁+d₂.
 7. The structure of claim 1, wherein the first and second layersare concentrically disposed with respect to each other in a directionparallel to a propagation direction of a guided wave.
 8. The structureof claim 1, wherein the first layer comprises said ENG material, saidENG material having a thickness d₁, an equivalent transverse impedanceX_(ENG), and an effective longitudinal wave number β_(ENG), and thesecond layer comprises said MNG material, said MNG material having athickness d₂, an equivalent transverse impedance X_(MNG), and aneffective longitudinal wave number β_(MNG), whereby the structureexhibits approximately zero reflection and tunneling if regions outsidesaid ENG and MNG materials are the same and if X_(ENG)=−X_(MNG) andβ_(ENG)d₁=β_(MNG)d₂.
 9. The structure of claim 8, wherein the firstlayer has a permittivity ε₁ and a permeability μ₁ and the second layerhas a permittivity ε₂ and a permeability μ₂, where ε₁=−ε₂ μ₁=μ₂, andd₁=d₂ whereby the structure exhibits zero-reflection and tunnelingthrough the first and second layers for any incident wave at any angleof incidence and for any polarization.
 10. A method of providing imagedisplacement of an object for near-field observation using the structureof claim 9, comprising the steps of placing the structure of claim 9between the object and an observer and observing evanescent waves thathave tunneled through the structure, whereby when the object andobserver are separated by a distance D, the object is displaced towardthe observer by the distance D−(d₁+d₂).